hydrodynamics Sir Horace Lamb ias long been the chief storehouse of information of all workers in hydrodynamics . . ." NATURL WF UNIVERSITY OF FLORIDA LIBRARIES ENGINEERING AND PHYSICS LI BRARY Digitized by the Internet Archive in 2013 http://archive.org/details/hydrodynamicsOOIamb HYDRODYNAMICS BY SIR HORACE LAMB, M.A, LL.D., Sc.D., F.R.S. HONORARY FELLOW OF TRINITY COLLEGE, CAMBRIDGE ; LATELY PROFESSOR OF MATHEMATICS IN THE VICTORIA UNIVERSITY OF MANCHESTER SIXTH EDITION NEW YORK DOVER PUBLICATIONS First Edition 1879 Second Edition 1895 Third Edition 1906 Fourth Edition 1916 Fifth Edition 1924 Sixth Edition 1932 FIRST AMERICAN EDITION 1945 BY SPECIAL ARRANGEMENT WITH CAMBRIDGE UNIVERSITY PRESS AND THE MACMILLAN CO. Library of Congress Catalog Card Number: 46-1891 Manufactured in the United States of America Dover Publications, Inc. 180 Varick Street New York 14, N. Y. PREFACE THIS may be regarded as the sixth edition of a Treatise on the Mathematical Theory of the Motion of Fluids, published in 1879. Subsequent editions, largely remodelled and extended, have appeared under the present title. In this issue no change has been made in the general plan and arrangement, but the work has again been revised throughout, some important omissions have been made good, and much new matter has been introduced. The subject has in recent years received considerable developments, in the theory of the tides for instance, and in various directions bearing on the problems of aeronautics, and it is interesting to note that the "classical" Hydrodynamics, often referred to with a shade of depreciation, is here found to have a widening field of practical applications. Owing to the elaborate nature of some of these researches it has not always been possible to fit an adequate account of them into the frame of this book, but attempts have occasionally been made to give some indication of the more important results, and of the methods employed. As in previous editions, pains have been taken to make due acknowledg- ment of authorities in the footnotes, but it appears necessary to add that the original proofs have often been considerably modified in the text. I have again to thank the staff of the University Press for much valued assistance during the printing. HORACE LAMB April 1932 CONTENTS CHAPTER I THE EQUATIONS OF MOTION ART. PAGE I, 2. Fundamental property of a fluid 1 3. The two plans of investigation ........ 1 4-9. ' Eulerian ' form of the equations *of motion. Dynamical equations. Equation of continuity.* Physical equations. Surface conditions . 2 10. Equation of energy 8 10 a. Transfer of momentum 10 II. Impulsive generation of motion 10 12. Equations referred to moving axes 12 13, 14. 'Lagrangian' form of the equations of motion and of the equation of continuity ........... 12 15, 16. Weber's transformation 14 16 a. Equations in polar co-ordinates 15 CHAPTER II INTEGRATION OF THE EQUATIONS IN SPECIAL CASES 17. Velocity -potential. Lagrange's theorem 17 18, 19. Physical and kinematical relations of (f> 18 20. Integration of the equations when a velocity-potential exists. Pressure- equation 19 21-23. Steady motion. Deduction of the pressure-equation from the principle of energy. Limiting velocity 20 24. Efflux of liquids ; vena contracta 23 24 a. 25. Efflux of gases 25 26-29. Examples of rotating fluid ; uniform rotation ; Rankine's ' combined vortex ' ; electromagnetic rotation 28 CHAPTER III IRROTATIONAL MOTION 30. Analysis of the differential motion of a fluid element into strain and rotation 31 31,32. . ' Flow ' and ' circulation.' Stokes' theorem 33 33. Constancy of circulation in a moving circuit . . . . . 35 34, 35. Irrotational motion in simply- connected spaces ; single- valued velocity- potential 37 viii Contents ART. PAGE 36-39. Incompressible fluids ; tubes of flow. <\> cannot be a maximum or mini- mum. The velocity cannot be a maximum. Mean value of $ over a spherical surface 38 40, 41. Conditions of determinateness of 41 42-46. Green's theorem ; dynamical interpretation ; formula for kinetic energy. Kelvin's theorem of minimum energy 43 47, 48. Multiply-connected regions ; ' circuits ' and ' barriers ' .... 49 49-51. Irrotational motion in multiply-connected spaces ; many- valued velocity- potential ; cyclic constants ........ 50 52. Case of incompressible fluids. Conditions of determinateness of <£ . . 53 53-55. Kelvin's extension of Green's theorem ; dynamical interpretation ; energy of an irrotationally moving liquid in a cyclic space .... 54 56-58. ' Sources ' and ' sinks ' ; double sources. Irrotational motion of a liquid in terms of surface-distributions of sources 57 CHAPTER IV MOTION OF A LIQUID IN TWO DIMENSIONS 59. Lagrange's stream -function .62 60. 60 a. Relations between stream- and velocity-functions. Two-dimensional sources. Electrical analogies ........ 63 61. Kinetic energy 66 62. Connection with the theory of the complex variable .... 66 63. 64. Simple types of motion, cyclic and acyclic. Image of a source in a circular barrier. Potential of a row of sources 68 65, 66. Inverse relations. Confocal curves. Flow from an open channel . . 72 67. General formulae ; Fourier method 75 68. Motion of a circular cylinder, without circulation ; stream-lines . . 76 69. Motion of a cylinder with circulation; 'lift.' Trochoidal path under a constant force 78 70. Note on more general problems. Transformation methods ; Kutta's problem 80 71. Inverse methods. Motion due to the translation of a cylinder; case of an elliptic section. Flow past an oblique lamina ; couple due to fluid pressure 83 72. Motion due to a rotating boundary. Rotating prismatic vessels of various sections. Rotating elliptic cylinder in infinite fluid ; general case with circulation 86 72 a. Representation of the effect at a distance of a moving cylinder by a double source . 90 72 b. Blasius' expressions for the forces on a fixed cylinder surrounded by an irrotationally moving liquid. Applications ; Joukowski's theorem ; forces due to a simple source 91 73. Free stream-lines. Schwarz' method of conformal transformation . . 94 74-78. Examples. Two-dimensional form of Borda's mouthpiece ; fluid issuing from a rectilinear aperture ; coefficient of contraction. Impact of a stream on a lamina, direct and oblique; resistance. Bobyleffs problem 96 79. Discontinuous motions . 105 80. Flow on a curved stratum 108 Contents ix CHAPTEE V IRROTATIONAL MOTION OF A LIQUID : PROBLEMS IN THREE DIMENSIONS ART. PAGE 81,82. Spherical harmonics. Maxwell's theory of poles 110 83. Laplace's equation in polar co-ordinates 112 84,85. Zonal harmonics. Hypergeometric series . 113 86. Tesseral and sectorial harmonics 116 87,88. Conjugate property of surface harmonics. Expansions . . . 118 89. Symbolical solutions of Laplace's equation. Definite integral forms . 119 90, 91. Hydrodynamical applications. Impulsive pressures over a spherical surface. Prescribed normal velocity. Energy of motion generated . 120 91 a. Examples. Collapse of a bubble. Expansion of a cavity due to internal pressure 122 92, 93. Motion of a sphere in an infinite liquid; inertia ' coefficient. Effect of a concentric rigid boundary . . . . . . . .123 94-96. Stokes' stream-function. Formulae in spherical harmonics. Stream-lines of a sphere. Images of a simple and a double source in a fiscal sphere. Forces on the sphere .125 97. Rankine's inverse method 130 98, 99. Motion of two spheres in a liquid. Kinematical formulae. Inertia coefficients 130 100, 101. Cylindrical harmonics. Solutions of Laplace's equation in terms of Bessel's functions. Expansion of an arbitrary function . . .134 102. Hydrodynamical examples. Flow through a circular aperture. Inertia coefficient of a circular disk . . . . . . . 137 103-106. Ellipsoidal harmonics for an ovary ellipsoid. Translation and rotation of an ovary ellipsoid in a liquid 139 107-109. Harmonics for a planetary ellipsoid. Flow through a circular aperture. Stream-lines of a circular disk. Translation and rotation of a planetary ellipsoid 142 110. Motion of a fluid in an ellipsoidal vessel 146 111. General orthogonal co-ordinates. Transformation of V 2 </> . . . 148 112. General ellipsoidal co-ordinates ; confocal quadrics 149 113. Flow through an elliptic aperture . . 150 114,115. Translation and rotation of an ellipsoid in liquid; inertia coefficients . 152 116. References to other problems 156 Appendix: The hydrodynamical equations referred to general ortho- gonal co-ordinates 156 CHAPTER VI ON THE MOTION OF SOLIDS THROUGH A LIQUID : DYNAMICAL THEORY 117,118. Kinematical formulae for the case of a single body 160 119. Theory of the 'impulse ' . 161 120. Dynamical equations relative to axes fixed in the body . . . .162 121, 121 a. Kinetic energy ; coefficients of inertia. Representation of the fluid motion at a distance by a double source 163 122, 123. Components of impulse. Reciprocal formulae 166 Contents ART. 124. 125. 126. 127- -129. 130. 131. 132- 134. 134 a. 135, 136. 137, 138. 139- -141. 142, 143. 144. PAGE Expressions for the hydrodynamic forces. The three permanent transla- tions ; stability 168 The possible modes of steady motion. Motion due to an impulsive couple 170 Types of hydrokinetic symmetry 172 Motion of a solid of revolution. Stability of motion parallel to the axis. Influence of rotation. Other types of steady motion . . .174 Motion of a ' helicoid ' 179 Inertia coefficients of a fluid contained in a rigid envelope . . . 180 Case of a perforated solid with cyclic motion through the apertures. Steady motion of a ring ; condition for stability . . . .180 The hydrodynamic forces on a cylinder moving in two dimensions . . 184 Lagrange's equations of motion in generalized co-ordinates. Hamiltonian principle. Adaptation to hydrodynamics .187 Examples. Motion of a sphere near a rigid boundary. Motion of two spheres in the line of centres 190 / Modification of Lagrange's equations in the case of cyclic motion ; ignoration of co-ordinates. Equations of a gyrostatic system . . 192 Kineto-statics. Hydrodynamic forces on a solid immersed in a non- uniform stream 197 Note on the intuitive extension of dynamic principles .... 201 CHAPTER VII 145. 146. 147. 148, 149. 150. 151. 152, 153. 154, 155. 156. 157. 158, 159. 159 i. 160. 161- 163. 164. 165. 166. 166 a. 167 VORTEX MOTION ' Vortex-lines ' and ' vortex-filaments ' ; kinematical properties . . 202 Persistence of vortices ; Kelvin's proof. Equations of Cauchy, Stokes, and Helmholtz. Motion in a fixed ellipsoidal envelope, with uniform vorticity ............ 203 Conditions of determinateness ......... 207 Velocity in terms of expansion and vorticity ; electromagnetic analogy. Velocities due to an isolated vortex . 208 Velocity-potential due to a vortex . .211 Vortex-sheets 212 Impulse and energy of a vortex-system ....... 214 Rectilinear vortices. Stream-lines of a vortex-pair. Other examples . 219 Investigation of the stability of a row of vortices, and of a double row. Karman's ' vortex-street ' 224 Kirchhoff's theorems on systems of parallel vortices .... 229 Stability of a columnar vortex of finite section ; Kirchhoff's elliptic vortex 230 Motion of a solid in a liquid of uniform vorticity 233 Vortices in a curved stratum of fluid 236 Circular vortices ; potential- and stream-function of an isolated circular vortex ; stream-lines. Impulse and energy. Velocity of translation of a vortex-ring 236 Mutual influence of vortex-rings. Image of a vortex-ring in a sphere . 242 General conditions for steady motion of a fluid. Cylindrical and spherical vortices 243 References 246 Bjerknes' theorems . 247 Clebsch's transformation of the hydrodynamical equations . . . 248 ART. 168. 169- -174. 175. 176. 177- -179. 180- -184. Contents xi CHAPTER VIII TIDAL WAVES PAGE General theory of small oscillations ; normal modes ; forced oscillations . 250 Free waves in uniform canal; effect of initial conditions; measuring of the approximations ; energy 254 Artifice of steady motion . . . . 261 Superposition of wave-systems ; reflection 262 Effect of disturbing forces ; free and forced oscillations in a finite canal . 263 Canal theory of the tides. Disturbing potentials. Tides in an equatorial canal, and in a canal parallel to the equator; semi-diurnal and diurnal tides. Canal coincident with a meridian ; change of mean level ; fortnightly tide. Equatorial canal of finite length ; lag of the tide 267 Waves in a canal of variable section. Examples of free and forced oscillations ; exaggeration of tides in shallow seas and estuaries . 273 Waves of finite amplitude ; change of type in a progressive wave. Tides of the second order 278 Wave motion in two horizontal dimensions ; general equations. Oscilla- tions of a rectangular basin ........ 282 Oscillations of a circular basin ; Bessel's functions ; contour lines. Elliptic basin ; approximation to slowest mode ...... 284 Case of variable depth. Circular basin 291 Propagation of disturbances from a centre ; Bessel's function of the second kind. Waves due to a local periodic pressure. General formula for diverging waves. Examples of a transient local disturbance . . 293 198-201. Oscillations of a spherical sheet of water ; free and forced waves. Effect of the mutual gravitation of the water. Reference to the case of a sea bounded by meridians and parallels 301 Equations of motion of a dynamical system referred to rotating axes . 307 Small oscillations of a rotating system ; stability 'ordinary' and 'secular.' Effect of a small degree of rotation on types and frequencies of normal modes 309 Approximate calculation of frequencies . . . . , . .313 Forced oscillations 316 Hydrodynamical examples ; tidal oscillations of a rotating plane sheet of water ; waves in a straight canal . . . . . . .317 Rotating circular basin of uniform depth ; free and forced oscillations . 320 Circular basin of variable depth 326 Examples of approximate procedure ....... 328 Tidal oscillations on a rotating globe. Laplace's kinetic theory . . 330 Symmetrical oscillations. Tides of long period 333 Diurnal and semi-diurnal tides. Discussion of Laplace's solution . . 340 Hough's investigations ; extracts and results 347 References to further researches 352 Modifications of the kinetic theory due to the actual configuration of the ocean ; question of phase . . 353 225, 226. Stability of the ocean. Remarks on the general theory of kinetic stability . 35£ Appendix : On Tide-generating Forces 358 185, 186. 187, 188. 189, 190. 191, 192. 193. 194- -197. 202, 203. 204- -205 a. 205 b. 206. 207, 208. 209- -211. 212. 212 a. 213, 214. 215- -217. 218- -221. 222, 223. 223, a. 224. xii Contents CHAPTER IX SURFACE WAVES ART. PAGE 227. The two-dimensional problem ; surface conditions 363 228. Standing waves ; lines of motion 364 229. 230. Progressive waves ; orbits of particles. Wave- velocity ; numerical tables. Energy of a simple-harmonic wave-train 366 231. Oscillations of superposed fluids 370 232. Instability of the boundary of two currents 373 233. 234. Artifice of steady motion 375 235. Waves in a heterogeneous liquid 378 236, 237. Group- velocity. Transmission of energy . 380 238-240. The Cauchy-Poisson wave-problem ; waves due to an initial local eleva- tion, or to a local impulse 384 241. Kelvin's approximate formula for the effect of a local disturbance in a linear medium. Graphical constructions 395 242-246. Surface-disturbance of a stream. Case of finite depth. Effect of inequali- ties in its bed 398 247. Waves due to a submerged cylinder 410 248, 249. General theory of waves due to a travelling disturbance. Wave- resistance 413 250. Waves of finite height ; waves of permanent type. Limiting form . . 417 251. Gerstner's rotational waves 421 252. 253. Solitary waves. Oscillatory waves of Korteweg and De Vries . . 423 254. Helmholtz' dynamical condition for waves of permanent type . . 427 255, 256. Wave-propagation in two horizontal dimensions. Effect of a local dis- turbance. Effect of a travelling pressure-disturbance; wave-patterns 429 256 a, 256 b. Travelling disturbances of other types. Ship-waves. Wave-resistance. Effect of finite depth on the wave-pattern 437 257-259. Standing waves in limited masses of water. Transverse oscillation in canals of triangular, and semi-circular section 440 260, 261. Longitudinal oscillations ; canal of triangular section ; edge- waves . 445 262-264. Oscillations of a liquid globe, lines of motion. Ocean of uniform depth on a spherical nucleus 450 265. Capillarity. Surface-condition 455 266. Capillary waves. Group-velocity 456 267. 268. Waves under gravity and capillarity. Minimum wave-velocity. Waves on the boundary of two currents 458 269. Waves due to a local disturbance. Effect of a travelling disturbance ; waves and ripples 462 270-272. Surface-disturbance of a stream ; formal investigation. Fish-line problem. Wave-patterns 464 273, 274 Vibrations of a cylindrical column of liquid. Instability of a jet . . 471 275 Oscillations of a liquid globe, and of a bubble 473 Contents xiii CHAPTER X WAVES OF EXPANSION ART. PAGE 276-280. Plane waves ; velocity of sound ; energy of a wave-system . . . 476 281-284. Plane waves of finite amplitude; methods of Riemann and Earnshaw. Condition for permanence of type ; Rankine's investigations. Waves of approximate discontinuity 481 285, 286. Spherical waves. Solution in terms of initial conditions . . . 489 287, 288. General equation of sound-waves. Equation of energy. Determinateness of solutions . 492 289. Simple-harmonic vibrations. Simple and double sources. Emission of energy ............ 496 290. Helmholtz' adaptation of Green's theorem. Velocity-potential in terms of surface-distributions of sources. Kirchhoff's formula . . . 498 291. Periodic disturbing forces 501 292. Applications of spherical harmonics. General formulae .... 503 293. Vibrations of air in a spherical vessel. Vibrations of a spherical stratum 506 294. Propagation of waves outwards from a spherical surface ; attenuation due to lateral motion 508 295. Influence of the air on the oscillations of a ball-pendulum ; correction for inertia ; damping . . . . . . . . . .510 296-298. Scattering of sound-waves by a spherical obstacle. Impact of waves on a movable sphere; case of synchronism . . . . . .511 299, 300. Diffraction when the wave-length is relatively large : by a flat disk, by an aperture in a plane screen, and by an obstacle of any form . 517 301. Solution of the equation of sound in spherical harmonics. Conditions at a wave-front ........... 521 302. Sound-waves in two dimensions. Effect of a transient source ; comparison with the one- and three-dimensional cases ..... 524 303. 304. Simple-harmonic vibrations ; solutions in Bessel functions. Oscillating cylinder. Scattering of waves by a cylindrical obstacle . . . 527 305. Approximate theory of diffraction of long waves in two dimensions. Diffraction by a flat blade, and by an aperture in a thin screen . 531 306,307. Reflection and transmission of sound-waves by a grating . . . 533 308. Diffraction by a semi-infinite screen ... .... 538 309, 310. Waves propagated vertically in the atmosphere; 'isothermal' and 'con- vective' hypotheses . . . .541 . 547 . 554 556 558 311, 311a, 312. Theory of long atmospheric waves 313. General equations of vibration of a gas under constant forces. 314, 315. Oscillations of an atmosphere on a non-rotating globe . 316. Atmosphere tides on a rotating globe. Possibility of resonance xiv Contents CHAPTER XI VISCOSITY ART. PAGE 317, 318. Theory of dissipative forces. One degree of freedom; free anu forced oscillations. Effect of friction on phase 562 319. Application to tides in equatorial canal ; tidal lag and tidal friction . 565 320. Equations of dissipative systems in general ; frictional and gyrostatic terms. Dissipation function 567 321. Oscillations of a dissipative system about a configuration of absolute equilibrium ' 568 322. Effect of gyrostatic terms. Example of two degrees of freedom ; dis- turbing forces of long period 570 323-325. Viscosity of fluids ; specification of stress ; formulae of transformation . 571 326, 327. The stresses as linear functions of rates of strain. Coefficient of viscosity. Boundary-conditions ; question of slipping 574 328. Dynamical equations. The modified Helmholtz equations; diffusion of vorticity - . 576 329. Dissipation of energy by viscosity 579 330, 330 a. Flow of a liquid between parallel planes. Hele Shaw's experiments. Theory of lubrication ; example 581 331, 332. Flow through a pipe of circular section; Poiseuille's laws; question of slipping. Other forms of section 585 333, 334. Cases of steady rotation. Practical limitations 587 334 a. Examples of variable motion. Diffusion of a vortex. Effect of surface- forces on deep water 590 335, 336. Slow steady motion ; general solution in spherical harmonics ; formulae for the stresses 594 337. Rectilinear motion of a sphere ; resistance ; terminal velocity ; stream- lines. Case of a liquid sphere ; and of a solid sphere, with slipping 597 338. Method of Stokes ; solutions in terms of the stream -function . . . 602 339. Steady motion of an ellipsoid 604 340. 341. Steady motion in a constant field of force 605 342. Steady motion of a sphere ; Oseen's criticism, and solution . . . 608 343. 343 a. Steady motion of a cylinder, treated by Oseen's method. References to other investigations 614 344. Dissipation of energy in steady motion; theorems of Helmholtz and Korteweg. Rayleigh's extension 617 345-347. Problems of periodic motion. Laminar motion, diffusion of vorticity. Oscillating plane. Periodic tidal force ; feeble influence of viscosity in rapid motions 619 348-351. Effect of viscosity on water-waves. Generation of waves by wind. Calming effect of oil on waves .......... 623 352, 353. Periodic motion with a spherical boundary ; general solution in spherical harmonics 632 354. Applications ; decay of motion in a spherical vessel ; torsional oscillations of a hollow sphere containing liquid 637 355. Effect of viscosity on the oscillations of a liquid globe .... 639 356. Effect on the rotational oscillations of a sphere, and on the vibrations of a pendulum 641 357. Notes on two-dimensional problems . 644 Contents xv ART. P AGE 358. Viscosity in gases ; dissipation function 645 359, 360. Damping of plane waves of sound by viscosity ; combined effect of viscosity and thermal conduction 646 360 a. Waves of permanent type, as affected by viscosity alone . . . . 650 360 b. Absorption of sound by porous bodies 652 361. Effect of viscosity on diverging waves 654 362, 363. Effect on the scattering of waves by a spherical obstacle, fixed or free . 657 364. Damping of sound-waves in a spherical vessel 661 365, 366. Turbulent motion. Reynolds' experiments ; critical velocities of water in a pipe ; law of resistance. Inferences from theory of dimensions 663 366 a. Motion between rotating cylinders 667 366 b. Coefficient of turbulence ; 'eddy' or 'molar' viscosity .... 668 366 c. Turbulence in the atmosphere ; variation of wind with height . . 669 367, 368. Theoretical investigations of Rayleigh and Kelvin 670 369. Statistical method of Reynolds 674 370. Resistance of fluids. Criticism of the discontinuous solutions of Kirchhoff and Rayleigh 678 370 a. Karman's formula for resistance 680 370 b. Lift due to circulation 681 371. Dimensional formulae. Relations between model and full-scale . . 682 371a, b, c. The boundary layer. Note on the theory of the aerofoil .... 684 37 Id, e, f, g. Influence of compressibility. Failure of stream-line flow at high speeds 691 CHAPTER XII ROTATING MASSES OF LIQUID 372. Forms of relative equilibrium. General theorems ..... 697 373. Formulae relating to attraction of ellipsoids. Potential energy of an ellipsoidal mass 700 374. Maclaurin's ellipsoids. Relations between eccentricity, angular velocity and angular momentum ; numerical tables 701 375. Jacobi's ellipsoids. Linear series of ellipsoidal forms of equilibrium. Numerical results 704 376. Other special forms of relative equilibrium. Rotating annulus . . 707 377. General problem of relative equilibrium ; Poineard's investigation. Linear series of equilibrium forms ; limiting forms and forms of bifurcation. Exchange of stabilities 710 378-380. Application to a rotating system. Secular stability of Maclaurin's and Jacobi's ellipsoids. The pear-shaped figure of equilibrium . . 713 381. Small oscillations of a rotating ellipsoidal mass; Poincar^'s method. References 717 382. Dirichlet's investigations; references. Finite gravitational oscillations of a liquid ellipsoid without rotation. Oscillations of a rotating ellipsoid of revolution 719 383. Dedekind's ellipsoid. The irrotational ellipsoid. Rotating elliptic cylinder 721 384. Free and forced oscillations of a rotating ellipsoidal shell containing liquid. Precession 724 385. Precession of a liquid ellipsoid 728 List of Authors cited 731 Index 734 HYDRODYNAMICS CHAPTER I THE EQUATIONS OF MOTION 1. The following investigations proceed on the assumption that the matter with which we deal may be treated as practically continuous and homogeneous in structure ; i.e. we assume that the properties of the smallest portions into which we can conceive it to be divided are the same as those of the substance in bulk. The fundamental property of a fluid is that it cannot be in equilibrium in a state of stress such that the mutual action between two adjacent parts is oblique to the common surface. This property is the basis of Hydrostatics, and is verified by the complete agreement of the deductions of that science with experiment. Very slight observation is enough, however, to convince us that oblique stresses may exist in fluids in motion. Let us suppose for instance that a vessel in the form of a circular cylinder, containing water (or other liquid), is made to rotate about its axis, which is vertical. If the angular velocity of the vessel be constant, the fluid is soon found to be rotat- ing with the vessel as one solid body. If the vessel be now brought to rest, the motion of the fluid continues for some time, but gradually subsides, and at length ceases altogether; and it is found that during this process the portions of fluid which are further from the axis lag behind those which are nearer, and have their motion more rapidly checked. These phenomena point to the existence of mutual actions between contiguous elements which are partly tangential to the common surface. For if the mutual action were everywhere wholly normal, it is obvious that the moment of momentum, about the axis of the vessel, of any portion of fluid bounded by a surface of revolution about this axis, would be constant. We infer, moreover, that these tangential stresses are not called into play so long as the fluid moves as a solid body, but only whilst a change of shape of some portion of the mass is going on, and that their tendency is to oppose this change of shape. 2. It is usual, however, in the first instance to neglect the tangential stresses altogether. Their effect is in many practical cases small, and, inde- pendently of this, it is convenient to divide the not inconsiderable difficulties of our subject by investigating first the effects of purely normal stress. The further consideration of the laws of tangential stress is accordingly deferred till Chapter XI. The Equations of Motion [chap. I If the stress exerted across any small plane area situate at a point P of the fluid be wholly normal, its intensity (per unit area) is the same for all aspects of the plane. The following proof of this theorem is given here for purposes of reference. Through P draw three straight lines PA, PB, PC mutually at right angles, and let a plane whose direction-cosines relatively to these lines are I, m, n, passing infinitely close to P, meet them in A, B, C. Let p, Pi, P2, Pz denote the intensities of the stresses* across the faces ABC, PBG, PC A, PAB, respectively, of the tetrahedron PABC. If A be the area of the first-mentioned face, the areas of the others are, in order, IA, mA, raA. Hence if we form the equation of motion of the tetrahedron parallel to PA we have p x . lA = pl . A, where we have omitted the terms which express the rate of change of momentum, and the component of the extraneous forces, because they are ultimately propor- tional to the mass of the tetrahedron, and therefore of the third order of small linear quantities, whilst the terms retained are of the second. We have then, ultimately, p—p\, and similarly p = p 2 = p 3 , which proves the theorem. 3. The equations of motion of a fluid have been obtained in two different forms, corresponding to the two ways in which the problem of determining the motion of a fluid mass, acted on by given forces and subject to given conditions, may be viewed. We may either regard as the object of our investigations a knowledge of the velocity, the pressure, and the density, at all points of space occupied by the fluid, for all instants; or we may seek to determine the history of every particle. The equations obtained on these two plans are conveniently designated, as by German mathematicians, the 'Eulerian' and the 'Lagrangian' forms of the hydrokinetic equations, although both forms are in reality due to Eulerf. The Eulerian Equations. 4. Let u, v, w be the components, parallel to the co-ordinate axes, of the velocity at the point (x, y, z) at the time t. These quantities are then functions of the independent variables x, y, z, t. For any particular value of t they define the motion at that instant at all points of space occupied by * Reckoned positive when pressures, negative when tensions. Most fluids are, however, incapable under ordinary conditions of supporting more than an exceedingly slight degree of tension, so that^ is nearly always positive. f " Principes generaux du mouvement des fluides," Hist, dc VAcad. dc Berlin, 1755. " De principiis motus fluidorum," Novi Comm. Acad. Petrop. xiv. 1 (1759). Lagrange gave three investigations of the equations of motion; first, incidentally, in 2-6] Eulerian Equations 3 the fluid; whilst for particular values of x, y, z they give the history of what goes on at a particular place. We shall suppose, for the most part, not only that u, v, w are finite and continuous functions of x, y, z, but that their space-derivatives of the first order (du/dx, dv/dx, dw/dx, &c.) are everywhere finite*; we shall understand by the term 'continuous motion,' a motion subject to these restrictions. Cases of exception, if they present themselves, will require separate examina- tion. In continuous motion, as thus defined, the relative velocity of any two neighbouring particles P, P' will always be infinitely small, so that the line PP' will always remain of the same order of magnitude. It follows that if we imagine a small closed surface to be drawn, surrounding P, and suppose it to move with the fluid, it will always enclose the same matter. And any surface whatever, which moves with the fluid, completely and permanently separates the matter on the two sides of it. 5. The values of u, v, w for successive values of t give as it were a series of pictures of consecutive stages of the motion, in which however there is no immediate means of tracing the identity of any one particle. To calculate the rate at which any function F (x, y, z, t) varies for a moving particle, we may remark that at the time t + 8t the particle which was originally in the position (x, ?/. z) is in the position (x + u8t, y + v8t, z + w8t), so that the corresponding value of F is F(x + u8t, y + v8t,z + iv8t, t + 8t) = F+u8t d -^ + v8t~- + w8t~ + 8t%- . 17 ox oy oz dt If, after Stokes, we introduce the symbol D/Dt to denote a differentiation following the motion of the fluid, the new value of F is also expressed by F+DF/Dt.8t, whence DF dF dF dF dF Bt = Tt+ U Tx + V dy + W dz ' (1) 6. To form the dynamical equations, let p be the pressure, p the density, X, T, Z the components of the extraneous forces per unit mass, at the point {x, y, z) at the time t. Let us take an element having its centre at (x, y, z), and its edges 8x, 8y, 8z parallel to the rectangular co-ordinate axes. The rate at which the ^-component of the momentum of this element is increasing is p8x8y8z DujDt; and this must be equal to the ^-component of the forces connection with the principle of Least Action, in the Miscellanea Taurinensia, ii. (1760) [Oeuvres, Paris, 1867-92, i.]; secondly in his "Memoire sur la Theorie du Mouvement des Fluides," Nouv. mem. de V Acad, de Berlin, 1781 [Oeuvres, iv.]; and thirdly in the Mecaniquc Analytique. In this last exposition he starts with the second form of the equations (Art. 14, below), but translates them at once into the ' Eulerian' notation. * It is important to bear in mind, with a view to some later developments under the head of Vortex Motion, that these derivatives need not be assumed to be continuous. 4 The Equations of Motion [chap, i acting on the element. Of these the extraneous forces give pBxByBzX. The pressure on the yz-fave which is nearest the origin will be ultimately that on the opposite face (p + \dp\dx . 8%) By Bz. The difference of these gives a resultant — dp/dx. BxByBz in the direction of ^-positive. The pressures on the remaining faces are perpendicular to x. We have then p Bx By Bz yc = pBxByBz X — ^-BxBy Bz. Substituting the value of DujDt from (1), and writing down the sym- metrical equations, we have du du du du _ Y 1 dp dt dx dy dz pdx' •(2) dv dv dv dv _ v 1 dp dt dx dy dz pdy' dw dw dw dw _ 7 1 dp dt dx dy dz p dz 7. To these dynamical equations we must join, in the first place, a certain kinematical relation between u, v, w, p, obtained as follows. If Q be the volume of a moving element, we have, on account of the constancy of mass, Dt \Dp 1 DQ . - P m + Qwr Q w To calculate the value of 1/Q .DQ/Dt, let the element in question be that which at time t fills the rectangular space BxByBz having one corner P at {%, y, z), and the edges PL, PM, PN (say) parallel to the co-ordinate axes. At time t + Bt the same element will form an oblique parallelepiped, and since the velocities of the particle L relative to the particle P are du/dx . Bx, dv/dx.Bx, dw/dx.Bx, the projections of the edge PL on the co-ordinate axes become, after the time Bt, (l+pSt)8*, d ^ti.Zx, d ^ St. Sec, \ dx ) dx dx respectively. To the first order in Bt, the length of this edge is now and similarly for the remaining edges. Since the angles of the parallelepiped * It is easily seen, by Taylor's theorem, that the mean pressure over any face of the element 5x by 5z may be taken to be equal to the pressure at the centre of that face. 6-t] Equation of Continuity 5 differ infinitely little from right angles, the volume is still given, to the first order in Bt, by the product of the three edges, i.e. we have 1 DQ dii dv dw (G> . or QDi = dx + dy + dz~ ( } Hence (1) becomes _s^®4; + S)=° ^ This is called the 'equation of continuity.' rvu - du dv dw //1X 1 he expression a" "*" a — ^~2~' ' ' which, as we have seen, measures the rate of dilatation of the fluid at the point (x,y,2), is conveniently called the 'expansion' at that point. From a more general point of view the expression (4) is called the 'divergence' of the vector (u,v,w); it is often denoted briefly by div (u, v, w). The preceding investigation is substantially that given by Euler*. Another, and now more usual, method of obtaining the equation of con- tinuity is, instead of following the motion of a fluid element, to fix the attention on an element BxByBz of space, and to calculate the change pro- duced in the included mass by the flux across the boundary. If the centre of the element be at (x, y, z), the amount of matter which per unit time enters it across the yz-f&ce nearest the origin is and the amount which leaves it by the opposite face is f pu + \ — '- — Bx j ByBz. BxByBz, The two faces together give a gain d .pu dx per unit time. Calculating in the same way the effect of the flux across the remaining faces, we have for the total gain of mass, per unit time, in the space BxByBz, the formula (d .pu 3 . pv d . pw\ j j j. Since the quantity of matter in any region can vary only in consequence of the flux across the boundary, this must be equal to ^(p BxByBz), * I.e. ante p. 2. 6 The Equations of Motion [chap, i whence we get the equation of continuity in the form ^ + 9 _£V_^ + ^ = (5) dt ox Oy oz v 8. It remains to put in evidence the physical properties of the fluid, so far as these affect the quantities which occur in our equations. In an 'incompressible' fluid, or liquid, we have Dp/Dt= 0, in which case the equation of continuity takes the simple form a-M4:=° • « It is not assumed here that the fluid is of uniform density, though this is of course by far the most important case. If we wish to take account of the slight compressibility of actual liquids, we shall have a relation of the form p = /e(p-po)lpo, (2) or plp = l+p//e, ..(3) where k denotes what is called the 'elasticity of volume.' In the case of a gas whose temperature is uniform and constant we have the ' isothermal ' relation PlPo = p/po> (4) where p , p are any pair of corresponding values for the temperature in question. In most cases of motion of gases, however, the temperature is not constant, but rises and falls, for each element, as the gas is compressed or rarefied. When the changes are so rapid that we can ignore the gain or loss of heat by an element due to conduction and radiation, we have the 'adiabatic' relation PlPo = (plpo) y , (5) where po and p are any pair of corresponding values for the element con- sidered. The constant 7 is the ratio of the two specific heats of the gas ; for atmospheric air, and some other gases, its value is about 1*408. 9. At the boundaries (if any) of the fluid, the equation of continuity is replaced by a special surface-condition. Thus at a fixed boundary, the velocity of the fluid perpendicular to the surface must be zero, i.e. if l> m, n be the direction-cosines of the normal, lu + mv + nw = (1) Again at a surface of discontinuity, i.e. a surface at which the values of u, v, w change abruptly as we pass from one side to the other, we must have l(ux — u 2 )-\-m (v 1 —v 2 )+ n(w 1 — w 2 ) = 0, (2) where the suffixes are used to distinguish the values on the two sides. The same relation must hold at the common surface of a fluid and a moving solid. 7-9] Boundary Condition 7 The general surface-condition, of which these are particular cases, is that if F(x, y, z, t) = be the equation of a bounding surface, we must have at every point of it DF/Dt = (3) For the velocity relative to the surface of a particle lying in it must be wholly tangential (or zero), otherwise we should have a finite flow of fluid across it. It follows that the instantaneous rate of variation of F for a surface-particle must be zero. A fuller proof, given by Lord Kelvin*, is as follows. To find the rate of motion (v) of the surface F(x, y, z, t)=0, normal to itself, we write F(x + lv8t, y + mv§t, z+nvdt, t + 8t) = 0, where I, m, n are the direction-cosines of the normal at (x, y y z). Hence 7 dF dF dF\ dF n Since (l,m,n)=^, ^, -^)+R, i dF wehave V= -R W (•*) At every point of the surface we must have v = lu + mv + nw, which leads, on substitution of the above values of I, m, n, %o the equation (3 N . The partial differential equation (3) is also satisfied by any surface moving with the fluid. This follows at once from the meaning of the operator DjDt. A question arises as to whether the converse necessarily holds ; i.e. whether a moving surface whose equation F=0 satisfies (3) will always consist of the same particles. Considering anv such surface let us fix our attention on a particle P situate on it at time t. The equation 3 expresses that the rate at which P is separating from the surface is at this instant zero : and it is easily seen that if the motion be continuous (according to the definition of Alt. 4\ the normal velocity, relative to the moving surface F, of a particle at an infinitesimal distance f from it is of the order f, viz. it is equal to G( where G is finite. Hence the equation of motion of the particle P relative to the surface may be written DUDt=GC. This shews that log £ increases at a finite rate, and since it is negative infinite to beo-in with (when £= u )> it remains so throughout, i.e. £ remains zero for the particle P. The same result follows from the nature of the solution of dF dF dF dF ct ox dy dz ' (o) considered as a partial differential equation in F\. The subsidiary system of ordinary differential equations is dt — — — dy _ d z u v ~~ w ' (6) * (W. Thomson) "Notes on Hydrodynamics," Camb. and Dub. Math. Journ. Feb 1848 [Mathematical and Physical Papers, Cambridge, 1882... , i. 83.] f Lagrange, Oeuvres, iv. 706. 8 The Equations of Motion [chap, i in which #, y, z are regarded as functions of the independent variable t. These are evidently the equations to find the paths of the particles, and their integrals may be supposed put in the forms x=fi(a,b,c,t), y=f 2 (a,b,c,t), z=f 3 (a, b, c, t), (7) where the arbitrary constants a, b, c are any three quantities serving to identify a particle ; for instance they may be the initial co-ordinates. The general solution of (5) is then found by elimination of a, b, c between (7) and F=yjr(a,b,c), (8) where \jr is an arbitrary function. This shews that a particle once in the surface F=0 remains in it throughout the motion. Equation of Energy. 10. In most cases which we shall have occasion to consider the extraneous forces have a potential; viz. we have X,Y,Z=-f,-f,-f (1) ox dy oz w The physical meaning of fl is that it denotes the potential energy, per unit mass, at the point (so, y, z), in respect of forces acting at a distance. It will be sufficient for the present to consider the case where the field of extraneous force is constant with respect to the time, i.e. 9f2/3£ = 0. If we now multiply the equations (2) of Art. 6 by u, v, w, in order, and add, we obtain a result which may be written If we multiply this by Sx By Sz, and integrate over any region, we find 5<r + 7)^///(.| + .| + .S^* <D where T=ifff p(u 2 + v 2 + w 2 )dxdydz ) V = fffClpdadydz, (3) i.e. T and V denote the kinetic energy and the potential energy in relation to the field of extraneous force, of the fluid which at the moment occupies the region in question. The triple integral on the right-hand side of (2) may be transformed by a process which will often recur in our subject. Thus, by a partial integration, llju^- dxdydz = 1 1 [pu] dydz — III p ~- dxdydz, where [pu] is used to indicate that the values of pu at the points where the boundary of the region is met by a line parallel to x are to be taken, with proper signs. If I, m, n be the direction-cosines of the inwardly directed normal to any element BS of this boundary, we have Sy8z = ± IBS, the signs alternating at the successive intersections referred to. We thus find that jf [pu] dydz = — ffpu I dS, 9-io] Energy 9 where the integration extends over the whole bounding surface. Transforming the remaining terms in a similar manner, we obtain ^ t {T + V)=j\ p{lu + m v + nw)dS + \\\p(^ + f y + d Qdxdydz. ...(4) In the case of an incompressible fluid this reduces to the form ^(T+V)=j[(lu + mv + nw)pdS. (5) Since lu + mv + nw denotes the velocity of a fluid particle in the direction of the normal, the latter integral expresses the rate at which the pressures pBS exerted from without on the various elements BS of the boundary are doing work. Hence the total increase of energy, kinetic and potential, of any portion of the liquid, is equal to the work done by the pressures on its surface. In particular, if the fluid be bounded on all sides by fixed walls, we have lu + mv + nw = over the boundary, and therefore T+F= const (6) A similar interpretation can be given to the more general equation (4), provided p be a function of p only. If we write E -/*<)■ (7) then E measures the work done by unit mass of the fluid against external pressure, as it passes, under the supposed relation between p and p, from its actual volume to some standard volume. For example, if the unit mass were enclosed in a cylinder with a sliding piston of area A, then when the piston is pushed outwards through a space B%, the work done is pA . Bx, of which the factor ABx denotes the increment of volume, i.e. of p~\ In the case of the adiabatic relation we find e= A_(p_pJ) (8) 7 - 1 \p pj We may call E the intrinsic energy of the fluid, per unit mass. Now, recalling the interpretation of the expression du/dx -f dv/dy + dwjdz, given in Art. 7, we see that the volume- integral in (4) measures the rate at which the various elements of the fluid are losing intrinsic energy by expansion; it is therefore equal to —DW/Dt, where W =/// Epdxdydz (9) Hence ~ (T + V+ W) = \\p(lu + mv + nw)dS (10) 10 The Equations of Motion [chap, i The total energy, which is now partly kinetic, partly potential in relation to a constant field of force, and partly intrinsic, is therefore increasing at a rate equal to that at which work is being done on the boundary by pressure from without. On the isothermal hypothesis we should have E = <?log(p/po), (11) where c 2 = p /p . This measures the 'free energy' per unit mass. With this definition of E we have an equation of the same form as (10), although the meaning is different. Transfer of Momentum. 10 a. If we fix our attention on the fluid which at the instant t occupies a certain region, the space which it occupies after a time St will differ from the original region by the addition of a surface film of (positive or negative) thickness (lu + mv -f nw) St, where (I, m, n) is the direction of the outward normal to the surface. Hence it is easy to see that the rate, at time t, at which the momentum of this particular portion of fluid is increasing is equal to the rate of increase of the momentum contained in a fixed region having the same boundary, together with the flux of momentum outwards across the boundary. In symbols, considering momentum parallel to Ox, we have Jjj g p dxdydz = jjj p g + u g + v g + w If) dxdydz p ^- dxdydz + \\ pu (lu + mv + nw) dS 'Zip") + d(pv) + dip\ s k ox 0y oz / pu dxdydz + \\ pu (lu + mv + mv)dS, (1) - \\\u . 0y _d L ~ dt by Art. 7 (5). In steady motion (Art. 21) the first term on the right hand disappears, and the rate of increase of momentum of any portion of fluid is equal to the flux of momentum outwards across its boundary. Conversely, if we apply the above principle to the fluid contained at any instant in a rectangular space SxSySz, we reproduce the equation of motion (Art. 6). Impulsive Generation of Motion. 11. If at any instant impulsive forces act bodily on the fluid, or if the boundary conditions suddenly change, a sudden alteration in the motion may take place. The latter case may arise, for instance, when a solid immersed in the fluid is suddenly set in motion. 10— ll] Generation of Motion 11 Let p be the density, u, v, w the component velocities immediately before, u', v r , vf those immediately after the impulse, X', Y', Z' the components of the extraneous impulsive forces per unit mass, m the impulsive pressure, at the point (x, y, z). The change of momentum parallel to x of the element defined in Art. 6 is then p8x8y8z(u — u); the ^-component of the extraneous impulsive forces is pSxSySzX' ', and the resultant impulsive pressure in the same direction is — dm/dx. SxSySz. Since an impulse is to be regarded as an infinitely great force acting for an infinitely short time (t, say), the effects of all finite forces during this interval are to be neglected. dm Hence, phxhyhz{u —u) — pSxSySzX' — ^— SxSySz, dx or Similarly, X v=Y'- w w = Z'- \dm_ p dx p dy ldvr p dz ■(i) These equations might also have been deduced from (2) of Art. 6, by multiplying the latter by Bt, integrating between the limits and r, putting ( T Zdt, vr={ T pdt, Jo Jo u — u = — V — — w w = — ■(2) X'=\ Xdt, F r = Ydt, Z' Jo Jo and then making r tend to the limit zero. In a liquid an instantaneous change of motion can be produced by the action of impulsive pressures only, even when no impulsive forces act bodily on the mass. In this case we have X', Y', Z' = 0, so thafc id™ p dx 19z? p d v Idja p dz If we differentiate these equations with respect to x, y, z y respectively, and add, and if we further suppose the density to be uniform, we find by Art. 8 (1) that dy* + dz* ~ ' The problem then, in any given case, is to determine a value of m satisfying this equation and the proper boundary conditions* ; the instantaneous change of motion is then given by (2). * It will appear in Chapter in. that the value of w is thus determinate, save as to an additive constant. 12 The Equations of Motion [chap, i Equations referred to Moving Axes. 12. It is sometimes convenient in special problems to employ a system of rectangular axes which is itself in motion. The motion of this frame may be specified by the component velocities u, v, w of the origin, and the com- ponent rotations p, q, r, all referred to the instantaneous positions of the axes. If u, v, w be the component velocities of a fluid particle at (x, y, z), the rates of change of its co-ordinates relative to the moving frame will be Dx Dy Dz j^ = u-u + ry-qz, j+=v-v + pz-rx, -^ =w- w + qa?-py. ...(1) After a time St the velocities of the particle parallel to the new positions of the co-ordinate axes will have become , (du du Dx du Dy du Dz\ ~ c - /ON To find the component accelerations we must resolve these parallel to the original positions of the axes in the manner explained in books on Dynamics. In this way we obtain the expressions (3) du du Dx du Dy du Dz dv dv Dx dv Dy dv Dz dt- VW+ru+ dxDi + dyWt + diDi dw dw Dx dw Du dw Dz __q M + p„ + __ + __ + ___ These will replace the expressions in the left-hand members of Art. 6 (2)*. The general equation of continuity is dp d ( Dx\ d ( Dy\ d ( Dz\ A reducing in the case of incompressibility to the form du dv dw _ „ dx dy dz ^ as before. The Lagrangian Equations. 13. Let a, b, c be the initial co-ordinates of any particle of fluid, x, y, z its co-ordinates at time t. We here consider x, y, z as functions of the independent variables a, b, c,t\ their values in terms of these quantities give the whole history of every particle of the fluid. The velocities parallel to * Greenhill, "On the General Motion of a Liquid Ellipsoid...," Proc. Gamb. Phil. Soc. iv. 4 (1880). i2-u] Lagrangian Equations 13 the axes of co-ordinates of the particle (a, b, e) at time t are dx/dt, dy/dt, dz/dt, and the component accelerations in the same directions are d 2 x/dt 2 , d 2 y/dt 2 , d 2 zjdt 2 . Let p be the pressure and p the density in the neighbourhood of this particle at time t; X, Y, Z the components of the extraneous forces per unit mass acting there. Considering the motion of the mass of fluid which at time t occupies the differential element of volume BwBySz, we find, by the same reasoning as in Art. 6, d 2 x _ Y 1 dp di 2 p dx ' dhj^ldp dt 2 p dy ' d 2 z _ „ _ 1 dp dt 2 ~ pdz' These equations contain differential coefficients with respect to oc, y, z, whereas our independent variables are a, b, c, t To eliminate these differential coef- ficients, we multiply the above equations by dx/da, dy/da, dzjda, respectively, and add; a second time by dx/db, dy/db, dz/db, and add; and again a third time by dxjdc, dy/dc, dz/dc, and add. We thus get the three equations 'c 2 x M 2 ' -*)£+©- ->)&♦©- J da pea 'd 2 x 3t 2 ' -*)S+@- ■*)&♦©- ■ z )i + "i=»' 'd 2 x jt 2 ' -*)«+©- -')&-©■ -z^+l*-o. jde pdc These are the 'Lagrangian' forms of the dynamical equations. 14. To find the form which the equation of continuity assumes in terms of our present variables, we consider the element of fluid which originally occupied a rectangular parallelepiped having its centre at the point (a, b, c), and its edges Sa, 8b, Be parallel to the axes. At the time t the same element forms an oblique parallelepiped. The centre now has for its co-ordinates x, y, z\ and the projections of the edges on the co-ordinate axes are respectively dx 5 dy * dz . fr- oa, TT'Oa, ^oa; oa oa oa !»■ >■ >■ £* i* s* 14 The Equations of Motion [chap, i The volume of the parallelepiped is therefore dx da dx db' dx do' da' dy db' dy dc' dz_ da dz_ db dz_ dc 8a 8b 8c, or, as it is often written, d (a, b, c) ■(1) (2) Hence, since the mass of the element is unchanged, we have d(x,y,z) _ P d(a~J^)- p0 > where p is the initial density at (a, b, c). In the case of an incompressible fluid p = p , so that (1) becomes d(x,y,z) ^ 1 d(a,b,c) Weber's Transformation. 15. If as in Art. 10 the forces X, Y, Z have a potential 12, the dynamical equations of Art. 13 may be written d 2 x dx d 2 y dy d 2 z dz _ 312 1 dp „ . dt 2 da dt 2 da dt 2 da~ da pda' Let us integrate these equations with respect to t between the limits and t. We remark that '* d 2 x dx o dt 2 da dx dx dt da dxdx . d dtda~ Uo ~ <i da~ f [*dx d 2 x , o Jo dtdadt [* (dx\ Jo \dt) dt, where u is the initial value of the ^-component of velocity of the particle (a, b, c). Hence if we write we find *4'[J?+ n -*®" + (IHI)*}]* » dx dx dy dy dz dz d% . dt da dt da dt da ° da dx dx dy dy dz dz dtdb + Jtdb + dtdb v = - dx dx dy dy dz dz _ didc' { "didc + dtdc~ Wo ~ db> dc' •(2) * H. Weber, "Ueber eine Transformation der hydrodynamischen Gleichungen," Crelle, lxviii. (1868). It is assumed in (1) that the density p, if not uniform, is a function of p only. 14-16 a] Polar Co-ordinates 15 These three equations, together with 3 I=I*--*{©' + (l)' + (l)'} ; <»> and the equation of continuity, are the partial differential equations to be satisfied by the five unknown quantities x,y, z,p>x'> P being supposed already eliminated by means of one of the relations of Art. 8. The initial conditions to be satisfied are x — a, y = b, z — c, % = 0. 16. It is to be remarked that the quantities a, b, c need not be restricted to mean the initial co-ordinates of a particle; they may be any three quanti- ties which serve to identify a particle, and which vary continuously from one particle to another. If we thus generalize the meanings of a, 6, c, the form of the dynamical equations of Art. 13 is not altered; to find the form which the equation of continuity assumes, let x ,y Q , z now denote the initial co-ordinates of the particle to which a, 6, c refer. The initial volume of the parallepiped, whose centre is at (x , y , z ) and whose edges correspond to variations Sa, Sb, Be of the parameters a, b, c, is d {yy°'*°h aSbSc, d (a, 6, c) sothatwehave p |£f*> = Po 3 -^£o) (1) r d (a, b, c) ru 3 (a, 6, c) v ; or, for an incompressible fluid, d(x t y,z) _ d(x 0y y ,z ) d(a,b,c) d(a,b,c) VW Equations in Polar Co-ordinates. 16 a. In the preceding investigations Cartesian co-ordinates have been employed, as is usually most consistent in the proof of general theorems. For special purposes polar co-ordinates are occasionally useful, and the appropriate formulae, on the 'Eulerian' plan, are accordingly given here for reference. In plane polars we may use u and v to denote the radial and transversal velocities, respectively, at the point (r, 6) at time t. Since the radius vector of a particle is revolving at the rate v/r, the ordinary theory of rotating axes gives for the component accelerations: Du v Dv v M-r- V < Di + r- U ' « where, by the method of Art. 5, I) d d d wrdt +u dr +v ri0 (2) The ' expansion ' (A) is found by calculating the rate of flux out of the quasi-rectangular element whose sides are Sr and rdd ; thus du u dv A + _ + (3) or r rod v ' 16 The Equations of Motion [chap, i In spherical polars we denote the radial velocity at (r, 6, cf>) by u, the velocity at right angles r in the plane of 6 by v, and the velocity at right angles to the plane of 6 by w. A triad of lines drawn from the origin parallel to these directors, when taken in this order, will, on the usual conventions, form a right-handed system. The changes in the angular co-ordinates of a particle in time dt are given by rdd = v8t, r sin 08(f> = wdt. This involves a rotation of the above system relative to its instantaneous position, with components cos 68(f), -sin<9Sc£, S<9. Hence if p, q, r are the components of the instantaneous angular velocity of the system, we have p = -cot0, q=--, r=- (4) The required accelerations of the particle which is at (/•, 6, (f>) are therefore .(5) Du Du v 2 -\-w 2 --n- + q W =-^ — , Dv Dv uv w 2 m - vw+ ru= m + 7 --cote, Dw Dio wu vw , , _-q % + p,= _- + _ + -coU where j- = k\+u ^+v-^ + io-—. — ^- (6) Dt dt or rdd rsmddcf* The expansion is found by calculating the flux out of the quasi-rectangular space whose edges are 8r, rb&, rsin 68cf>, and is du n u dv v , . dw /h ,. dr r rdd r rsm&dcfi CHAPTER II INTEGRATION OF THE EQUATIONS IN SPECIAL CASES 17. In a large and important class of cases the component velocities u, v, w can be expressed in terms of a single- valued function <£, as follows: d<b deb d<b /1X * «,„,«—£ - Ty , -f z (i)* Such a function is called a 'velocity-potential,' from its analogy with the potential function which occurs in the theories of Attractions, Electro- statics, &c. The general theory of the velocity-potential is reserved for the next chapter; but we give at once a proof of the following important theorem : If a velocity potential exist, at any one instant, for any finite portion of a perfect fluid in motion under the action of forces which have a potential, then, provided the density of the fluid be either constant or a function of the pressure only, a velocity-potential exists for the same portion of the fluid at all instants before or after f. In the equations of Art. 15, let the instant at which the velocity- potential (f>Q exists be taken as the origin of time; we have then u da + v db + w dc = — d<f> , throughout the portion of the mass in question. Multiplying the equations (2) of Art. 15 in order by da, db, dc, and adding, we get ^idx+ ^dy+~-dz — (u da + v db -\-w dc)——dx, or, in the 'Eulerian' notation, udx + vdy + wdz = — d ($ + %) = — d<p, say. Since the upper limit of t in Art. 15 (1) may be positive or negative, this proves the theorem. It is to be particularly noticed that this continued existence of a velocity- potential is predicated, not of regions of space, but of portions of matter. * The reasons for the introduction of the minus sign are stated in the Preface. The theory of ' cyclic ' velocity- potentials is discussed later. t Lagrange, "Memoire sur la Theorie du Mouvement des Fluides," Nouv. mem. de VAcad. de Berlin, 1781 [Oeuvres, iv. 714]. The argument is reproduced in the Mecanique Analytique. Lagrange's statement and proof were alike imperfect ; the first rigorous demonstration is due to Cauchy, "Memoire sur la Theorie des Ondes," Mem. de VAcad. roy. des Sciences, i. (1827) [Oeuvres Completes, Paris, 1882... , l re Serie, i. 38]; the date of the memoir is 1815. Another proof is given by Stokes, Camb. Trans, viii. (1845) (see also Math, and Phys. Papers, Cambridge, 1880... , i. 106, 158, and ii. 36), together with an excellent historical and critical account of the whole matter. 18 Integration of the Equations in Special Cases [chap, ii A portion of matter for which a velocity-potential exists moves about and carries this property with it, but the part of space which it originally occupied may, in the course of time, come to be occupied by matter which did not originally possess the property, and which therefore cannot have acquired it. The class of cases in which a single- valued velocity-potential exists includes all those where the motion has originated from rest under the action of forces of the kind here supposed ; for then we have, initially, u da + v db + w dc = 0, or cj) = const. The restrictions under which the above theorem has been proved must be carefully remembered. It is assumed not only that the extraneous forces X, Y, Z, estimated at per unit mass, have a potential, but that the density p is either uniform or a function of p only. The latter condition is violated, for example, in the case of the convection currents generated by the unequal application of heat to a fluid; and again, in the wave-motion of a hetero- geneous but incompressible fluid arranged originally in horizontal layers of equal density. Another case of exception is that of 'electro-magnetic rotations'; see Art. 29. 18. A comparison of the formulae (1) with the equations (2) of Art. 11 leads to a simple physical interpretation of (j>. Any actual state of motion of a liquid, for which a (single-valued) velocity-potential exists, could be produced instantaneously from rest by the application of a properly chosen system of impulsive pressures. This is evident from the equations cited, which shew, moreover, that (f> — vr/p + const. ; so that ot = p</> + C gives the requisite system. In the same way ts = — p$ + C gives the system of impulsive pressures which would completely stop the motion*. The occurrence of an arbitrary constant in these expressions merely shews that a pressure uniform throughout a liquid mass produces no effect on the motion. In the case of a gas, (f> may be interpreted as the potential of the extraneous impulsive forces by which the actual motion at any instant could be produced instantaneously from rest. A state of motion for which a velocity-potential does not exist cannot be generated or destroyed by the action of impulsive pressures, or of extraneous impulsive forces having a potential. 19. The existence of a velocity-potential indicates, besides, certain kine- matical properties of the motion. A 'line of motion' is denned to be a line drawn from point to point, so * This interpretation was given by Canchy, loc, cit., and by Poisson, Mem. de VAcad. roy. des Science*, i. (1816). 17-20] Velocity-Potential 19 that its direction is everywhere that of the motion of the fluid. The diffe- rential equations of the system of such lines are dx dy dz ,_. — = — = — (2) U V w The relations (1) shew that when a velocity-potential exists the lines of motion are everywhere perpendicular to a system of surfaces, viz. the 'equi- potential 5 surfaces <£ = const. Again, if from the point (x, y, z) we draw a linear element 8s in the direction (I, m, n), the velocity resolved in this direction is lu + mv + nw, or dcj> dx dcj>dy d(j>dz , . , _ d<j> dx ds dy ds dz ds' ds' The velocity in any direction is therefore equal to the rate of decrease of cf) in that direction. Taking 8s in the direction of the normal to the surface $ = const., we see that if a series of such surfaces be drawn corresponding to equidistant values of 0, the common difference being infinitely small, the velocity at any point will be inversely proportional to the distance between two consecutive surfaces in the neighbourhood of the point. Hence, if any equipotential surface intersect itself, the velocity is zero at the intersection. The intersection of two distinct equipotential surfaces would imply an infinite velocity. 20. Under the circumstances stated in Art. 17, the equations of motion are at once integrable throughout that portion of the fluid mass for which a velocity-potential exists provided p is either constant, or a definite function of p. For in virtue of the relations dv/dz = dw/dy, dw/dx = du/dz, du/dy = dv/dx, which are implied in (1), the equations of Art. 6 may be written d 2 cj> du dv dw 311 19p dxdt ox dx dx dx pox These have the integral l + W + E = f t +F(t) (4) Here q denotes the resultant velocity (u 2 + v 2 -f w 2 )%, F(t) is an arbitrary function of t, and E is defined by Art. 10 (7), and has (in the case of a gas) the interpretation there given. Our equations take a specially simple form in the case of an incompressible fluid; viz. we then have MS-a-w+m :■■«> 20 Integration of the Equations in Special Cases [chap, ii with the equation of continuity w + df + M- 0> (6) which is the equivalent of Art. 1 (8). When, as in many cases which we shall have to consider, the boundary conditions are purely kinematical, the process of solution consists in finding a function which shall satisfy (5) and the prescribed surface-conditions. The pressure p is then given by (4), and is thus far indeterminate to the extent of an additive function of t. It becomes determinate when the value of p at some point of the fluid is given for all values of t. Since the term F{t) is without influence on resultant pressures it is frequently omitted. Suppose, for example, that we have a solid or solids moving through a liquid com- pletely enclosed by fixed boundaries, and that it is possible {e.g. by means of a piston) to apply an arbitrary pressure at some point of the boundary. Whatever variations are made in the magnitude of the force applied to the piston, the motion of the fluid and of the solids will be absolutely unaffected, the pressure at all points instantaneously rising or falling by equal amounts. Physically, the origin of the paradox (such as it is) is that the fluid is treated as absolutely incompressible. In actual liquids changes of pressure are propagated with very great, but not infinite, velocity. If the co-ordinate axes are in motion, the formula for the pressure is -»('*-3H('B-a-'('&-'i9 <" where q 2 = (u - u) 2 + (v — v) 2 4- (w — w) 2 (8) This easily follows from the formulae for the accelerations given in Art. 12 (3). Steady Motion. 21. When at every point the velocity is constant in magnitude and direction, i.e. when ^ = ^ = ^ = (1) dt ' dt v ' dt ' w everywhere, the motion is said to be 'steady/ In steady motion the lines of motion coincide with the paths of the particles. For if P, Q be two consecutive points on a line of motion, a particle which is at any instant at P is moving in the direction of the tangent at P, and will, therefore, after an infinitely short time arrive at Q. The motion being steady, the lines of motion remain the same. Hence the direction of motion at Q is along the tangent to the same line of motion, i.e. the particle continues to describe the line, which is now appropriately called a 'stream-line.' 20-22] Steady Motion 21 The stream-lines drawn through an infinitesimal contour define a tube, which may be called a 'stream-tube.' In steady motion the equations (3) of Art. 20 give / ^ = _n-ig 2 + constant (2) The law of variation of pressure along a stream-line can however in this case be found without assuming the existence of a velocity-potential. For if 8s denote an element of a stream-line, the acceleration in the direction of motion is qdq/ds, and we have dq_ dn ldp , q ds--fo-~pds-> W whence integrating along the stream-line, p = -il-W + C. (4) This is similar in form to (2), but is more general in that it does not assume the existence of a velocity-potential. It must however be carefully noticed that the 'constant' of equation (2) and the 'C of equation (4) have different meanings, the former being an absolute constant, while the latter is constant along any particular stream-line, but may vary as we pass from one stream- line to another. 22. The theorem (4) stands in close relation to the principle of energy. If this be assumed independently, the formula may be deduced as follows*. Taking first the particular case of a liquid, consider the filament of fluid which at a given instant occupies a length AB of a stream-tube, the direction of motion being from A to B. Let p be the pressure, q the velocity, fl the potential of the extraneous forces, a- the area of the cross-section, at A> and let the values of the same quantities at B be distinguished by accents. After a short interval of time the filament will occupy a length A-iBx, let m be the mass included between the cross-sections at A and A lf or B and B±. Since the motion is steady, the gain of energy by the filament will be Again, the net work done on it is pm/p — p'm/p. Equating the increment of energy to the work done, we have p p or, using in the same sense as before, £ Cl-tf+C, (5) P which is what the equation (4) becomes when p is constant. * This is really a reversion to the methods of Daniel Bernoulli, Hydrodynamica, Argentorati, 1738. 22 Integration of the Equations in Special Cases [chap, ii To prove the corresponding formula for compressible fluids, we remark that the fluid crossing any section has now, in addition to its energies of motion and position, the energy ('intrinsic' or 'free' as the case may be) -K)--M? : per unit mass. The addition of these terms in (5) gives the equation (4). In the case of a gas subject to the adiabatic law PlPo = (p/po) y , (6) the equation (4) takes the form i _£_£__, Wg . + a (7) 23. The preceding equations shew that, in steady motion, and for points along any one stream-line*, the pressure is, cceteris paribus, greatest where the velocity is least, and vice versa. This statement becomes evident when we reflect that a particle passing from a place of higher to one of lower pressure must have its motion accelerated, and vice versdf. It follows that in any case to which the equations of the last Article apply there is a limit which the velocity cannot exceed J. For instance, let us suppose that we have a liquid flowing from a reservoir where the velocity may be neglected, and the pressure is p , and that we may neglect extraneous forces. We have then, in (5), G = po/p, and therefore P=Po~ipq 2 (8) Now although it is found that a liquid from which all traces of air or other dissolved gas have been eliminated can sustain a negative pressure, or tension, of considerable magnitude §, this is not the case with fluids such as we find them under ordinary conditions. Practically, then, the equation (8) shews that q cannot exceed (2p /p)%. This limiting velocity is that with which the fluid would escape from the reservoir into a vacuum. In the case of water at atmospheric pressure it is the velocity * due to ' the height of the water- barometer, or about 45 feet per second. If in any case of fluid motion of which we have succeeded in obtaining the analytical expression, we suppose the motion to be gradually accelerated until the velocity at some point reaches the limit here indicated, a cavity will be formed there, and the conditions of the problem are more or less changed. It will be shewn, in the next chapter (Art. 44), that in irrotational motion of a liquid, whether 'steady' or not, the place of least pressure is always at * It will be shewn later that this restriction is unnecessary when a velocity-potential exists. t Some interesting practical illustrations of this principle are given by Froude, Nature, xiii. 1875. X Cf. Helmholtz, "Ueber discontinuirliche Flussigkeitsbewegungen, " Berl, Monatsber. April 1868; Phil. Mag. Nov. 1868 [Wissenschaftliche Abhandlungen, Leipzig, 1882-3, i. 146]. 0. Reynolds, Manch. Mem. vi. (1877) [Scientific Papers, Cambridge, 1900... , i. 231]. 22-24] Steady Motion 23 some point of the boundary, provided the extraneous forces have a potential XI satisfying the equation d 2 a dHi d 2 n_ dx 1 dy 2 dz 2 This includes, of course, the case of gravity. In the general case of a fluid in which p is a given function of p we have, putting H = 0, q = 0, in (4), -if* W Jv 9 P P For a gas subject to the adiabatic law, this gives y-1 v2 _ 2 7 Po iS 1 -©' w flr- 7 - 1 p ( \p / = ^l(Co 2 -c 2 ), (11) if c, = (yp/p)%, = (dp/dp)i, denote the velocity of sound in the gas when at pressure p and density p, and c the corresponding velocity for gas under the conditions which obtain in the reservoir. (See Chapter x.) Hence the limiting velocity is 2 Co, or 2-214c , if 7 = 1*408. t^y 24. We conclude this chapter with a few simple applications of the equations. Flow of Liquids, Let us take in the first instance the problem of the efflux of a liquid from a small orifice in the walls of a vessel which is kept filled up to a constant level, so that that motion may be regarded as steady. The origin being taken in the upper surface, let the axis of z be vertical, and its positive direction downwards, so that Q, — — gz. If we suppose the area of the upper surface large compared with that of the orifice, the velocity at the former may be neglected. Hence, determining the value of C in Art. 21 (4) so that p = P (the atmospheric pressure) when z = 0, we have* Z-Z+gg-W (1) r r At the surface of the issuing jet we have p = P, and therefore <? 2 = 2<7*, ■ (2) i.e. the velocity is that due to the depth below the upper surface. This is known as Torricelli s Theorem^. * This result is due to D. Bernoulli, I.e. ante p. 21. f "De motu gravium naturaliter aceeierato," Firenze, 1643. 24 Integration of the Equations in Special Cases [chap, ii We cannot however at once apply this result to calculate the rate of efflux of the fluid, for two reasons. In the first place, the issuing fluid must be regarded as made up of a great number of elementary streams converging from all sides towards the orifice. Its motion is not, therefore, throughout the area of the orifice, everywhere perpendicular to this area, but becomes more and more oblique as we pass from the centre to the sides. Again, the converging motion of the elementary streams must make the pressure at the orifice somewhat greater in the interior of the jet than at the surface, where it is equal to the atmospheric pressure. The velocity, therefore, in the interior of the jet will be somewhat less than that given by (2). Experiment shews however that the converging motion above spoken of ceases at a short distance beyond the orifice, and that (in the case of a circular orifice) the jel, then becomes approximately cylindrical. The ratio of the area of the section S r of the jet at this point (called the 'vena contracta') to the area $ of the orifice is called the 'coefficient of contraction.' If the orifice be simply a hole in a thin wall, this coefficient is found experimentally to be about '62. The paths of the particles at the vena contracta being nearly straight, there is little or no variation of pressure as we pass from the axis to the outer surface of the jet. We may therefore assume the velocity there to be uniform throughout the section, and to have the value given by (2), where z now denotes the depth of the vena contracta below the surface of the liquid in the vessel. The rate of efflux is therefore Vg^.pS' (3) The calculation of the form of the issuing jet presents difficulties which have only been overcome in a few ideal cases of motion in two dimensions. (See Chapter iv.) It may however be shewn that the coefficient of con- traction must, in general, lie bet wen h and 1. To put the argument in its simplest form, let us first take the case of liquid issuing from a vessel the pressure in which, at a distance from the orifice, exceeds that in the external space by the amount P, gravity being neglected. When the orifice is closed by a plate, the resultant pressure of the fluid on the containing vessel is of course nil. If when the plate is removed we assume (for the moment) that the pressure on the walls remains sensibly equal to P, there will be an un- balanced pressure PS acting on the vessel in the direction opposite to that of the jet, and tending to make it recoil. The equal and contrary reaction on the fluid produces in unit time the velocity q in the mass pqS' flowing through the 'Vena contracta,' whence PS = pq 2 8' (4) The principle of energy gives, as in Art. 22, p=w (5) 24-24 a] Vena Contracta 25 so that, comparing, we have S' — \S. The formula (1) shews that the pressure on the walls, especially in the neighbourhood of the orifice, will in reality fall somewhat below the static pressure P, so that the left-hand side of (4) is an under-estimate. The ratio S'/S will therefore in general be > J. In one particular case, viz. where a short cylindrical tube, projecting inwards, is attached to the orifice, the assumption above made is sufficiently exact, and the consequent value ^ for the coefficient then agrees with experiment. The reasoning is easily modified so as to take account of gravity (or other conservative forces). We have only to substitute for P the excess of the static pressure at the level of the orifice over the pressure outside. The difference of level between the orifice and the ' vena contracta' is here neglected*. Another important application of Bernoulli's theorem is to the measure- ment of the velocity of a stream by means of a 'Pitot tube.' This consists of a fine tube open at one end, which points up-stream, and connected at the other end with a manometer. Along the stream-line which is in a line with the axis of the tube the velocity falls rapidly from q to 0, so that the manometer indicates the value of the 'total head' p + ^pq 2 in the neighbourhood. A second manometer connected with a tube closed at the end, but with minute perfora- tions in the wall, past which the stream glides, determines the value of the 'static pressure' p. The density p being known, a comparison of the readings gives the value of q. The two contrivances are often combined in one instru- ment. The method is extensively used in Aerodynamics, the compressibility of the air being found to have little effect up to speeds of the order of 200 ft. per sec. Floiv of a Gas. 2A&. The steady now of a gas subject to the adiabatic law presents some features of interest. Let a be the cross-section at any point of a stream -tube, and hs an element of the length in the direction of flow. Omitting extraneous forces we have in place of Art. 23 (10) *-*- M HSl <■» * The above theory is due to Borda (Mem. de VAcad. des Sciences, 1766), who also made experiments with the special form of mouth-piece referred to, and found S/S' = 1-942. It was re-discovered by Hanlon, Proc. Lond. Math. Soc. iii. 4 (1869) ; the question is further elucidated in a note appended to this paper by Maxwell. See also Froude and J. Thomson, Proc. Glasgotv Phil. Soc. x. (1876). It has been remarked by several writers that in the case of a diverging conical mouth-piece projecting inwards the section at the vena contracta may be less than half the area of the internal orifice. 26 Integration of the Equations in Special Cases [chap, ii where the zero suffix relates to some fixed section of the tube. If c be the velocity of sound corresponding to the local values of p and p this may be written ? 2 + 2 2 3 7— 1 X 7— 1 Again, since the mass crossing any section in unit time is the same, pqa- = p q (To (3) .p. 1 da _ 1 dq 1 dp dp jnence 7 — ~~ 7 ~n ~j a- as qds pap as --£(-$) « It follows from (2) and (4) that in a converging tube q will increase and c diminish, or vice versa, according as q is less or greater than c. For a diverging tube the statements must be reversed. Briefly, we may say that in a converging tube the stream velocity and the local velocity of sound continually approach one another, whilst in a diverging tube they separate more and more. These results follow also from a graphical representation of the equations (2) and (3). Since c 2 is proportional to p? ~ 1 , the latter may be written 2 2 cy- 1 qo- = c y- 1 q o- (5) If we take abscissae proportional to c and ordinates to q, the equation (2) represents an ellipse of invariable shape, drawn through the point (c , q ). For any assigned value of ct/o-q the equation (5) represents a sort of hyperbolic curve. For a certain value (</) of o- this will touch the ellipse, and we then have q = c. The curves A A', BB\ CO' in the annexed diagram correspond to the ratios cr cr 7 = 8, 4, 2, respectively, whilst the point D corresponds to the minimum section a. For still smaller values of o- the intersections with the ellipse are imaginary, and steady adiabatic flow becomes impossible. The diagram shews that for any section greater than a there are two possible pairs of values of q and c, as has been remarked by Osborne Reynolds and others. When q is less than c the representative point on the ellipse lies below OB. In a converging tube it assumes a sequence of positions such as A\ B', C\ the stream-velocity increasing, and the velocity of sound decreasing, as the critical section a' is approached. When q is greater than c, on the other hand, the repre- sentative point lies above OD. In a converging tube we have a sequence such as A, B, C ; the stream-velocity decreases, and the velocity of sound increases. 24 a-25] Flow of Gases 27 25. We consider more particularly the efflux of a gas, supposed to flow through a small orifice from a vessel in which the pressure is p and density po into a space where the pressure is p x . If the ratio po/Pi of the pressure inside and outside the vessel do not exceed a certain limit, to be indicated presently, the flow will take place in much the same manner as in the case of a liquid, and the rate of discharge may be found by putting p=pi in Art. 23 (10), and multiplying the resulting value of q by the area o-i of the vena contracta. This gives for the rate of discharge of mass* y+i »-&M®'-®7~ .(6) It is plain however that there must be a limit to the applicability of this result ; for otherwise we should be led to the paradoxical conclusion that when pi=0, i.e. the discharge is into a vacuum, the flux of matter is nil. The elucidation of this point is due to Prof Osborne Reynolds f. It appears that qp is a maximum, i.e. the section of an elementary stream is a minimum, when a.s appears from (4) the velocity of the stream is equal to tht velocity of sound in gas of the pressure and density which prevail there. On the adiabatic hypothesis this gives, by Art. 23 (11), *-(£)* <" -«-*- i-i^-r- tiyht <»> or, if y= 1-408, p = -634p , p=-527p (9) If pi be less than this value, the stream after passing the point in question widens out again, until it is lost at a distance in the eddies due to viscosity. The minimum sections of the elementary streams will be situate in the neighbourhood of the orifice, and their sum S may be called the virtual area of the latter. The velocity of efflux, as found from (2), is q=-9Uc . The rate of discharge is then = qpS, where q and p have the values just found, and is therefore approximately independent of the external pressure p x so long as this falls below 527p . The physical reason of this is (as pointed out by Reynolds) that, so long as the velocity at any point exceeds the velocity of sound under the conditions which obtain there, no change of pressure can be propagated backwards beyond this point so as to affect the motion higher up the stream J. Some recent experiments of Stanton § confirm in all essentials the views of Reynolds, and clear up some apparent discrepancies. Under similar circumstances as to pressure, the velocities of efflux of different gases are (so far as y can be assumed to have the same value for each) proportional to the corresponding velocities of sound. Hence (as we shall see in Chapter x.) the velocity of efflux will vary inversely, and the rate of discharge of mass will vary directly, as the square root of the density ||. * A result equivalent to this was given by Saint Venant and Wantzel, Journ. de VEcole Polyt. xvi. 92 (1839), and was discussed by Stokes, Brit. Ass. Reports for 1846 [Papers, i. 176]. f "On the Flow of Gases," Proc. Manch. Lit. and Phil. Soc. Nov. 17, 1885; Phil. Mag. March 1886 [Papers, ii. 311]. A similar explanation was given by Hugoniot, Comptes Rendus, June 28, July 26, and Dec. 13, 1886. X For a further discussion and references see Rayleigh, "On the Discharge of Gases under High Pressures," Phil. Mag. (6) xxxii. 177 (1916) [Scientific Papers, Cambridge, 1899-1920, vi. 407]. § Proc. Roy. Soc. A, cxi. 306 (1926). || Cf. Graham, Phil. Trans. 1846. 28 Integration of the Equations in Special Cases [chap, ii Rotating Liquid. 26. Let us next take the case of a mass of liquid rotating, under the action of gravity only, with constant and uniform angular velocity co about the axis of z, supposed drawn vertically upwards. By hypothesis, u, v, w = — coy, cox, 0, X, Y,Z= 0, 0, -g. The equation of continuity is satisfied identically, and the dynamical equations obviously are pdx a pdy pdz These have the common integral ^- = \co 2 (x 2 + y 2 )-gz + const (2) P The free surface, p = const., is therefore a paraboloid of revolution about the axis of z, having its concavity upwards, and its latus rectum = 2g/co 2 . «. dv du Since - — = 2 co, dx cy a velocity-potential does not exist. A motion of this kind could not therefore be generated in a 'perfect' fluid, i.e. in one unable to sustain tangential stress. 27. Instead of supposing the angular velocity co to be uniform, let us suppose it to be a function of the distance r from the axis, and let us inquire what form must be assigned to this function in order that a velocity-potential may exist for the motion. We find dv du _ 9 dco dx dy dr ' and in order that this may vanish we must have cor 2 — /x, a constant. The velocity at any point is then = /n/r, so that the equation (2) of Art. 21 becomes ^ = const.-i^, (1) p 2 T l V } if no extraneous forces act. To find the value of <£ we have, using polar co-ordinates, dr ' rdO r y whence cp= — fiO + const. = — /jl tan -1 - + const (2) x We have here an instance of a 'cyclic' function. A function is said to be 'single-valued' throughout any region of space when we can assign to every point of that region a definite value of the function in such a way that these values shall form a continuous system. This is not possible with the function 26-28] Rotating Liquid 29 (2) ; for the value of 6, if it vary continuously, changes by — 2irfjb as the point to which it refers describes a complete circuit round the origin. The general theory of cyclic velocity-potentials will be given in the next chapter. If gravity act, and if the axis of z be drawn vertically upwards, we must add to (1) the term — gz. The form of the free surface is therefore that generated by the revolution of the hyperbolic curve a?z = const, about the axis of z. By properly fitting together the two preceding solutions we obtain the case of Rankine's ' combined vortex.' Thus the motion being everywhere in coaxial circles, let us suppose the velocity to be equal to <or from r = to r==a, and to o*a 2 /r for r>a. The corresponding- forms of the free surface are then given by and these being continuous with one another when r=a. The depth of the central depression below the general level of the surface is therefore oo 2 a 2 /g. 28. To illustrate, by way of contrast, the case of extraneous forces not having a potential, let us suppose that a mass of liquid filling a right circular cylinder moves from rest under the action of the forces X = Ax+By, Y=B'x + Cy, Z=0, the axis of z being that of the cylinder. If we assume u— — coy, v = a>x, w = 0, where o> is a function of t only, these values satisfy the equation of continuity and the boundary conditions. The dynamical equations are evidently <2r= A <r. -X- Tin — , dco ~di (d*x=Ax +By , 2 y = B'x+Cy- ■(1) 1 dp dt ™ * ~ " " ' " u p dx ' j Differentiating the first of these with respect to y, and the second with respect to x, and subtracting, we eliminate p, and find %~U*-B). ■(2) 30 Integration of the Equations in Special Cases [chap, ii The fluid therefore rotates as a whole about the axis of z with constantly accelerated angular velocity, except in the particular case when B = B'. To find p, we substitute the value of datjdt in (1) and integrate ; we thus get p^ where 2p=B + B'. £o> 2 (x 2 +y 2 ) -f £ {Ax 2 + 2$xy + Cf) + const., 29. As a final example, we will take one suggested by the theory of 'electro-magnetic rotations.' If an electric current be made to pass radially from an axial wire, through a conducting liquid, to the walls of a metallic containing cylinder, in a uniform magnetic field, the extraneous forces will be of the type* Assuming u= — a>y, v=<ox, w = 0, where to is a function of r and t only, we have 00) „ uX 1 0» 3£ ^ r 2 p dy Eliminating p, we obtain 2 -=- + r ^-^- = 0. The solution of this is a> = F (t)/r 2 +f(r), where F and /denote arbitrary functions. If o> = when t=0, we have F(0)lr 2 +f(r)=0, and therefore o> = „ — — = -* , •(2) where X is a function of t which vanishes for £=0. Substituting in (1), and integrating, we find H'-tK'l-^+xw Since p is essentially a single-valued function, we must have d\jdt=^ or X=p,t. Hence the fluid rotates with an angular velocity which varies inversely as the square of the distance from the axis, and increases constantly with the time. If C denote the total flux of electricity outwards, per unit length of the axis, and y the component of the magnetic force t parallel to the axis, we have u = yCI2irp. The above case is specially simple, in that the forces X, Y, Z have a potential (£1= -a tan -1 yjx), though a 'cyclic' one. As a rule, in electro-magnetic rotations, this is not the case. CHAPTER III IRROTATIONAL MOTION 30. The present chapter is devoted mainly to an exposition of some general theorems relating to the kinds of motion already considered in Arts. 17-20; viz. those in which udx+ vdy -\-wdz is an exact differential throughout a finite mass of fluid. It is convenient to begin with the following analysis, due to Stokes*, of the motion of a fluid element in the most general case. The component velocities at the point (x, y, z) being u, v, w, the relative velocities at an infinitely near point (x + Bx, y + By, z + 8z) are du * , du z du* dx dz S v = d ^8x + d ^Sy + d £8z, dx If we write a dw g, dw * dw* dx dy u dz du ''dx' .(1) _P_dw dv S~dy + dz' _ dw dv dy dz' 9 7 dv dy dw dz du dw dz dx ' , dv du dx dy' du dz div dx £-£. ?= dv du dx dy t. (2) equations (1) may be written Bu — aBx + ^hBy + \gBz + \{rjBz—t > By)/ Bv = ihBx+ b8y + if8z + i(Z8x-%8z), - (3) 8w = | gBx -\- if By + cBz + \{%8y — rjBx). Hence the motion of a small element having the point (x, y, z) for its centre may be conceived as made up of three parts. * " On the Theories of the Internal Friction of Fluids in Motion, &c." Camb. Phil. Trans. viii. (1845) [Papers, i, 80]. t There is here a deviation from the traditional convention. It has been customary to use symbols such as £, 17, £* (Helmholtz) or w', w", a/" (Stokes) to denote the component rotations 1 /dw dv\ T/du dw\ 1 fdv du\ Zydy'TzJ' 2\dz~dx)' 2\dx~dy) of a fluid element. The fundamental kinematical theorem is however that of Art. 32 (3), and the definition of If, y, f adopted in the text avoids the intrusion of an unnecessary factor 2 (or \ as the case may be) in this and in a whole series of subsequent formulae relating to vortex motion. It also improves the electro-magnetic analogy of Art. 148. 32 Irrotational Motion [chap, hi The first part, whose components are u, v, w, is a motion of translation of the element as a whole. The second part, expressed by the first three terms on the right-hand sides of the equations (3), is a motion such that, if 8x, 8y y 8z be regarded as current co-ordinates, every point is moving in the direction of the normal to that quadric of the system a {hxf + b (8yf + c (8zf +f8y 8z + g8z8x + h8x8y = const. . . .(4) on which it lies. If we refer these quadrics to their principal axes, the cor- responding parts of the velocities parallel to these axes will be 8u' = a'8x', hv' = Vty\ oV = c'S/, (5) if a' (oV) 2 + V (8y') 2 + c' {8z'f = const. is what (4) becomes by the transformation. The formulae (5) express that the length of every line in the element parallel to x' is being elongated at the (positive or negative) rate a\ whilst lines parallel to y' and z' are being- elongated in like manner at the rates b' and c' respectively. Such a motion is called one of pure strain and the principal axes of the quadrics (4) are called the axes of the strain. The last two terms on the right-hand sides of the equations (3) express a rotation of the element as a whole about an instantaneous axis; the component angular velocities of the rotation being Jf, ^rj, Jf*. The vector whose components are £, 77, f may conveniently be called the 'vorticity' of the medium at the point (x, y, z). This analysis may be illustrated by the so-called 'laminar' motion of a liquid. Thus if u = /jLy, y=0, w = 0, we have a, b, c,f, g, £, 77 = 0, h = fx, (= -/*. If A represent a rectangular fluid element bounded by planes parallel to the co-ordinate planes, then B represents the change produced in this in a short time by the strain alone, and C that due to the strain plus the rotation. It is easily seen that the above resolution of the motion is unique. If we assume that the motion relative to the point (x, y, z) can be made up of a strain and a rotation in which the axes and coefficients of the strain and the axis and angular velocity of the rotation are arbitrary, then calculating the * The quantities corresponding to ££, £77, £f in the theory of the infinitely small displacements of a continuous medium had been interpreted by Cauchy as expressing the ' mean rotations ' of an element, Exercices d' Analyse et de Physique, ii. 302 (Paris, 1841). 3o-3i] Deformation of an Element 33 relative velocities Bu, Bv, Bw, we get expressions similar to those on the right- hand sides of (3), but with arbitrary values of a, b, c,f, g, h, f, 77, ? Equating coefficients of Bx, By, Bz, however, we find that a, b, c, &c. must have respectively the same values as before. Hence the directions of the axes of the strain, the rates of extension or contraction along them, and the axis and the amount of the vorticity, at any point of the fluid, depend only on the state of relative motion at that point, and not on the position of the axis of reference. When throughout a finite portion of a fluid mass we have f , rj, f all zero, the relative motion of any element of that portion consists of a pure strain only, and is called 'irrotational.' 31. The value of the integral J(udx + vdy + wdz), [( dx dy dz\ 7 \{ U ds +V Ts* W ds) ds > taken along any line ABCD, is called* the 'flow' of the fluid from A to D along that line. We shall denote it for shortness by I (ABCD). If A and D coincide, so that the line forms a closed curve, or circuit, the value of the integral is called the 'circulation' in that circuit. We denote it by I (ABC A). If in either case the integration be taken in the opposite direction, the signs of dx/ds, dy/ds, dzjds will be reversed, so that we have I(AD) = -I(DA), and I (ABCA) = - 1 (ACBA). It is also plain that I (ABCD) = I (AB) + 1 (BC) 4- 1 (CD). Again, any surface may be divided, by a double series of lines crossing it, into infinitely small elements. The sum of the circulations round the boundaries of these elements, taken all in the same sense, is equal to the circulation round the original boundary of the surface (supposed for the moment to consist of a single closed curve). For, in the sum in question, the flow along each side common to two elements comes in twice, once for each element, but with opposite signs, and there- fore disappears from the result. There remain then only the flows along those sides which are parts of the original boundary; whence the truth of the above statement. From this it follows, by considerations of continuity, that the circulation round the boundary of any surface-element BS, having a given position and aspect, is ultimately proportional to the area of the element. * Sir W. Thomson, " On Vortex Motion," Edin. Trans, xxv. (1869) [Papers, iv. 13]. 34 Irrotational Motion [chap, hi If the element be a rectangle 8y8z having its centre at the point (a, y, z\ then calculating the circulation round it in the direction shewn by the arrows in the annexed figure, we have y \ i B ! p ! A ~ B 1 A z I{AB) = I(CD) = . _i (dv/dz)8z}8y, I (BC) [w + i(dw/dy)8y}8z, [w-%(dw/dy)8y}8z, and therefore [v + i(dv/dz)8z}8y, I (DA)-- In this way we infer that the circulations round the boundaries of any infinitely small areas 8Si, S$ 2 , 8S 3 , having their planes parallel to the co-ordinate planes, are fSSi, v8S 2) £8S 3 , (1) respectively. Again, referring to the figure and the notation of Art. 2, we have I {ABC A) = I (PBGP) + / (PGAP) + / (PABP) = £ JA + ?; . 77* A + f . nA, whence we infer that the circulation round the boundary of any infinitely small area 8S is (Zf + mi7 + n£)8& (2) We have here an independent proof that the quantities f, tj, f, as defined by Art. 30 (2), may be regarded as the components of a vector. It will be observed that some convention is implied as to the relation between the sense in which the circulation round the boundary of 8S is estimated, and the sense of the normal (I, m, n). In order to have a clear understanding on this point, we shall suppose in this book that the axes of co-ordinates form a right-handed system ; thus if the axes of x and y point E. and N. respectively, that of z will point vertically upwards*. The sense in * Maxwell, Proc. Lond. Math. Soc. (1) iii. 279, 280. Thus in the above diagram the axis of x is supposed drawn towards the reader. 31-33] Circulation in a Finite Circuit 35 which the circulation, as given by (2), is estimated is then related to the direction of the normal (I, m y n) in the manner typified by a right-handed screw*. 32. Expressing now that the circulation round the edge of any finite surface is equal to the sum of the circulations round the boundaries of the infinitely small elements into which the surface may be divided, we have, by (2), l(udx + vdy + wdz) = tf(l% + m'n + nZ)dS, (3) or, substituting the values of f , 77, f from Art. 30, i(^ + ^ + ^)=//f(|4:) +ro s-£) + »g-|)}^... w where the single-integral is taken along the bounding curve, and the double- integral over the surface f. In these formulae the quantities I, m, n are the direction-cosines of the normal drawn always on one side of the surface, which we may term the positive side; the direction of integration in the first member is then that in which a man walking on the surface, on the positive side of it, and close bo the edge, must proceed so as to have the surface always on his left hand. The theorem (3) or (4) may evidently be extended to a surface whose boundary consists of two or more closed curves, provided the integration in the first member be taken round each of these in the proper direction, according to the rule just given. ^^^^^^^^^^ Thus, if the surface-integral in (4) extend over the shaded portion of the annexed figure, the directions in which the circulations in the several parts of the boundary are to be taken are shewn by the arrows, the positive side of the surface being that which faces the reader. The value of the surface-integral taken over a closed surface is zero. It should be noticed that (4) is a theorem of pure mathematics, and is true whatever functions u, v, w may be of %, y, z, provided only they be con- tinuous and differentiable at all points of the surface J. 33. The rest of this chapter is devoted to a study of the kinematical properties of irrotational motion in general, as defined by the equations fc%t=0, (1) * See Maxwell, Electricity and Magnetism, Oxford, 1873, Art. 23. t This theorem is due to Stokes, Smith's Prize Examination Papers for 1854. The first pub- lished proof appears to have been given by Hankel, Zur allgem. Theorie der Bewegung der Flussigkeiten, Gottingen, 1861. That given above is due to Lord Kelvin, I.e. ante p. 33. See also Thomson and Tait, Natural Philosophy, Art. 190 (j), and Maxwell, Electricity and Magnetism, Art. 24. X It is not necessary that their differential coefficients should be continuous. 36 Irrotational Motion [chap, hi i.e. the circulation in every infinitely small circuit is assumed to be zero. The existence and properties of the velocity-potential in the various cases that may arise will appear as consequences of this definition. The physical importance of the subject rests on the fact that if the motion of any portion of a fluid mass be irrotational at any one instant it will under certain very general conditions continue to be irrotational. Practically, as will be seen, this has already been established by Lagrange's theorem, proved in Art. 17, but the importance of the matter warrants a repetition of the investigation, in terms of the Eulerian notation, in the form given by Lord Kelvin*. Consider first any terminated line A B drawn in the fluid, and suppose every point of this line to move always with the velocity of the fluid at that point. Let us calculate the rate at which the flow along this line, from A to B, is increasing. If 8x, 8y, 8z be the projections on the co-ordinate axes of an element of the line, we have D , k v Da . D8x Now D8x/Dt, the rate at which 8x is increasing in consequence of the motion of the fluid, is equal to the difference of the velocities parallel to x at the two ends of the element, i.e. to 8u\ and the value of Bu/Dt is given by Art. 5. Hence, and by similar considerations, we find, if p be a function of p only, and if the extraneous forces X, Y, Z have a potential Q, T -(u8x + v8y + w8z) = — - — SO + u8u + v8v + w8w. JJt p Integrating along the line, from A to B, we get D [ B -y- (udx + vdy + wdz) = r ■(2) DtJ A or the rate at which the flow from A to B is increasing is equal to the excess of the value which — Jdp/p — H + \(f has at B over that which it has at A. This theorem comprehends the whole of the dynamics of a perfect fluid. For instance, equations (2) of Art. 15 may be derived from it by taking as the line AB the infinitely short line whose projections were originally 8a, 8b, 8c, and equating separately to zero the coefficients of these infinitesimals. If fl be single-valued, the expression within brackets on the right-hand side of (2) is a single-valued function of x, y, z. Hence if the integration on the left-hand side be taken round a closed curve, so that B coincides with A, we have D f jr I (udx + vdy + wdz) = 0, (3) or, the circulation in any circuit moving with the fluid does not alter with the time. * I.e. ante p. 33. 33-35] Velocity-Potential 37 It follows that if the motion of any portion of a fluid mass be initially irrotational it will always retain this property; for otherwise the circulation in every infinitely small circuit would not continue to be zero, as it is initially by virtue of Art. 32 (3). 34. Considering now any region occupied by irrotationally- moving fluid, we see from Art. 32 (3) that the circulation is zero in every circuit which can be filled up by a continuous surface lying wholly in the region, or which in other words is capable of being contracted to a point without passing out of the region. Such a circuit is said to be 'reducible.' Again, let us consider two paths A GB, ADB, connecting two points A, B of the region, and such that either may by continuous variation be made to coincide with the other, without ever passing out of the region. Such paths are called 'mutually reconcileable.' Since the circuit AGBDA is reducible, we have I (ACBDA)=0, or since I (BDA) = -I(ADB), I(ACB) = I(ADB); i.e. the flow is the same along any two reconcileable paths. A region such that all paths joining any two points of it are mutually reconcileable is said to be 'simply-connected.' Such a region is that enclosed within a sphere, or that included between two concentric spheres. In what follows, as far as Art. 46, we contemplate only simply-connected regions. 35. The irrotational motion of a fluid within a simply-connected region is characterized by the existence of a single-valued velocity-potential. Let us denote by — (f> the flow to a variable point P from some fixed point A, viz. </> = — I (udx + vdy + wdz) (1) J A The value of (p has been shewn to be independent of the path along which the integration is effected, provided it lie wholly within the region. Hence </> is a single-valued function of the position of P; let us suppose it expressed in terms of the co-ordinates (a, y, z) of that point. By displacing P through an infinitely short space parallel to each of the axes of co-ordinates in succession, we find u-- d -$ v-- d -& w-- d -i (2) i.e. cj> is a velocity-potential, according to the definition of Art. 17. The substitution of any other point B for A, as the lower limit of the integral in (1), simply adds an arbitrary constant to the value of $, viz. the flow from A to B. The original definition of cj> in Art. 17, and the physical interpretation in Art. 18, alike leave the function indeterminate to the extent of an additive constant. 38 Irrotational Motion [chap, hi As we follow the course of any line of motion the value of <£ continually decreases; hence in a simply-connected region the lines of motion cannot form closed curves. 36. The function cf> with which we have here to do is, together with its first differential coefficients, by the nature of the case, finite, continuous, and single-valued at all points of the region considered. In the case of incom- pressible fluids, which we now proceed to consider more particularly, <£ must also satisfy the equation of continuity, (6) of Art. 20, or as we shall in future write it, for shortness, V^ = 0, (1) at every point of the region. Hence </> is now subject to mathematical conditions identical with those satisfied by the potential of masses attracting or repelling according to the law of the inverse square of the distance, at all points external to such masses; so that many of the results proved in the theories of Attractions, Electrostatics, Magnetism, and the Steady Flow of Heat, have also a hydrodynamical application. We proceed to develop those which are most important from this point of view. In any case of motion of an incompressible fluid the surface-integral of the normal velocity taken over any surface, open or closed, is conveniently called the 'flux' across the surface. It is of course equal to the volume of fluid crossing the surface per unit time. When the motion is irrotational, the flux is given by -IK* where BS is an element of the surface, and 8n an element of the normal to it, drawn in the proper direction. In any region occupied wholly by liquid, the total flux across the boundary is zero, i.e. // a,"**- . < 2 > the element Sn of the normal being drawn always on one side (say inwards), and the integration extending over the whole boundary. This may be regarded as a generalized form of the equation of continuity (1). The lines of motion drawn through the various points of an infinitesimal circuit define a tube, which may be called a tube of flow. The product of the velocity (q) into the cross-section (a, say) is the same at all points of such a tube. We may, if we choose, regard the whole space occupied by the fluid as made up of tubes of flow, and suppose the size of the tubes so adjusted that the product qa is the same for each. The flux across any surface is then proportional to the number of tubes which cross it. If the surface be closed, 35-38] Tubes of Flow 39 the equation (2) expresses the fact that as many tubes cross the surface inwards as outwards. Hence a line of motion cannot begin or end at a point internal to the fluid. 37. The function $ cannot be a maximum or a minimum at a point in the interior of the fluid ; for, if it were, we should have d<f>/dn everywhere positive, or everywhere negative, over a small closed surface surrounding the point in question. Either of these suppositions is inconsistent with (2). Further, the square of the velocity cannot be a maximum at a point in the interior of the fluid. For let the axis of x be taken parallel to the direction of the velocity at any point P. The equation (1), and therefore also the equation (2), is satisfied if we write d<f)/d% for <£. The above argument then shews that d<f>/dx cannot be a maximum or a minimum at P. Hence there must be points in the immediate neighbourhood of P at which (d(f>/dx) 2 and therefore a fortiori is greater than the square of the velocity at P *. On the other hand, the square of the velocity may be a minimum at some point of the fluid. The simplest case is that of a zero velocity ; see, for example, the figure of Art. 69, below. 38. Let us apply (2) to the boundary of a finite spherical portion of the liquid. If r denote the distance of any point from the centre of the sphere, Sot the elementary oolid angle subtended at the centre by an element SS of the surface, we have d(pldn = — d(f)/dr, and &S = r 2 8sx. Omitting the factor r 2 , (2) becomes n , d«r=0, or d or ajj**'- 00 Since 1/4jt . ffad-n or I/4nrr 2 .jf<f>dS measures the mean value of <f> over the surface of the sphere, (3) shews that this mean value is independent of the radius. It is therefore the same for any sphere, concentric with the former one, which can be made to coincide with it by gradual variation of the radius, without ever passing out of the region occupied by the irrotationally moving liquid. We may therefore suppose the sphere contracted to a point, and so obtain a simple proof of the theorem, first given by Gauss in his * This theorem was enunciated, in another connection, by Lord Kelvin, Phil. Mag. Oct. 1850 [Reprint of Papers on Electrostatics, dtc, London, 1872, Art. 665]. The above demonstration is due to Kirchhoff, Vorlesungen iiber mathematische Physik, Mechanik, Leipzig, 1876. For another proof see Art. 44 below. 40 Irrotational Motion [chap, hi memoir* on the theory of Attractions, that the mean value of cj> over any spherical surface throughout the interior of which (1) is satisfied, is equal to its value at the centre. The theorem, proved in Art. 37, that <f> cannot be a maximum or a minimum at a point in the interior of the fluid, is an obvious consequence of the above. The above proof appears to be due, in principle, to Frost f. Another demon- stration, somewhat different in form, was given by the late Lord RayleighJ. The equation (1), being linear, will be satisfied by the arithmetic mean of any number of separate solutions fa, fa, fa, — Let us suppose an infinite number of systems of rectangular axes to be arranged uniformly about any point P as origin, and let fa, fa, fa, ... be the velocity-potentials of motions which are the same with respect to these several systems as the original motion </> is with respect to the system x, y, z. In this case the arithmetic mean (<£, say) of the functions fa, fa, fa, ... will be a function of r, the distance from P, only. Expressing that in the motion (if any) represented by </>, the flux across any spherical surface which can be contracted to a, point, without passing out of the region occupied by the fluid, would be zero, we have dr or (j) = const. 39. Again, let us suppose that the region occupied by the irrotationally moving fluid is f periphractic,'§ i.e. that it is limited internally by one or more closed surfaces, and let us apply (2) to the space included between one (or more) of these internal boundaries, and a spherical surface completely enclosing it (or them) and lying wholly in the fluid. If M denote the total flux into this region, across the internal boundary, we find, with the same notation as before, 80 I! dr the surface-integral extending over the sphere only. This may be written M csll^- 47T? -hence 4^//^ = lr \\^ = £~r + ° (4) * " Allgemeine Lehrsatze, u.s.w.," Resultate aus den Beobachtungcn des magnetischenVereins, 1839 [Werke, Gottingen, 1870-80, v. 199]. f Quarterly Journal of Mathematics, xii. (1873). X Messenger of Mathematics, vii. 69 (1878) [Papers, i. 347]. § See Maxwell, Electricity and Magnetism, Arts. 18, 22. A region is said to be ' aperiphractic ' wlren every closed surface drawn in it can be contracted to a point without passing out of the region. 38-4o] Mean Value over a Spherical Surface 41 That is, the mean value of <£ over any spherical surface drawn under the above-mentioned conditions is equal to Mj^mrr -I- G, where r is the radius, M an absolute constant, and G a quantity which is independent of the radius but may vary with the position of the centre*. If however the original region throughout which the irrotational motion holds be unlimited externally, and if the first derivative (and therefore all the higher derivatives) of c/> vanish at infinity, then G is the same for all spherical surfaces enclosing the whole of the internal boundaries. For if such a sphere be displaced parallel to xf, without alteration of size, the rate at which C varies in consequence of this displacement is, by (4), equal to the mean value of d(j>/dx over the surface. Since dcp/dx vanishes at infinity, we can by taking the sphere large enough make the latter mean value as small as we please. Hence C is not altered by a displacement of the centre of the sphere parallel to x. In the same way we see that G is not altered by a displacement parallel to y or z\ i.e. it is absolutely constant. If the internal boundaries of the region considered be such that the total flux across them is zero, e.g. if they be the surfaces of solids, or of portions of incompressible fluid whose motion is rotational, we have M = 0, so that the mean value of (/> over any spherical surface enclosing them all is the same. 40. (a) If cf> be constant over the boundary of any simply-connected region occupied by liquid moving irrotationally, it has the same constant value throughout the interior of that region. For if not constant it would necessarily have a maximum or a minimum value at some point of the region. Otherwise: we have seen in Arts. 35, 36 that the lines of motion cannot begin or end at any point of the region, and that they cannot form closed curves lying wholly within it. They must therefore traverse the region, beginning and ending on its boundary. In our case however this is impossible, for such a line always proceeds from places where </> is greater to places where it is less. Hence there can be no motion, i.e. dcf> d<j> d(f) _ « dx ' dy ' dz and therefore cj> is constant and equal to its value at the boundary. (/3) Again, if d</>/dn be zero at every point of the boundary of such a region as is above described, <j> will be constant throughout the interior. For the condition d<p/dn = expresses that no lines of motion enter or leave the region, but that they are all contained within it. This is however, as we have seen, inconsistent with the other conditions which the lines must conform to. Hence, as before, there can be no motion, and cj> is constant. * It is understood, of course, that the spherical surfaces to which this statement applies are reconcileable with one another, in a sense analogous to that of Art. 34. f Kirchhoff, Mechanik, p. 191. 42 Irrotational Motion [chap, hi This theorem may be otherwise stated as follows: no continuous irrota- tional motion of a liquid can take place in a simply-connected region bounded entirely by fixed rigid walls. (7) Again, let the boundary of the region considered consist partly of surfaces S over which <f> has a given constant value, and partly of other surfaces 2 over which d<f>/dn = 0. By the previous argument, no lines of motion can pass from one point to another of S, and none can cross S. Hence no such lines exist ; <p> is therefore constant as before, and equal to its value at 8. It follows from these theorems that the irrotational motion of a liquid in a simply-connected region is determined when either the value of <f>, or the value of the inward normal velocity — d<f>/dn, is prescribed at all points of the boundary, or (again) when the value of cf> is given over part of the boundary, and the value of — d<j>/dn over the remainder. For if fa , fa be the velocity- potentials of two motions each of which satisfies the prescribed boundary- conditions, in any one of these cases, the function fa — fa satisfies the condition (a) or (/3) or (7) of the present Article, and must therefore be constant throughout the region. 41. A class of cases of great importance, but not strictly included in the scope of the foregoing theorems, occurs when the region occupied by the irrotationally moving liquid extends to infinity, but is bounded internally by one or more closed surfaces. We assume, for the present, that this region is simply-connected, and that (j> is therefore single-valued. If cj> be constant over the internal boundary of the region, and tend every- where to the same constant value at an infinite distance from the internal boundary, it is constant throughout the region. For otherwise <£ would be a maximum or a minimum at some point within the region. We infer, exactly as in Art. 40, that if <f> be given arbitrarily over the internal boundary, and have a given constant value at infinity, its value is everywhere determinate. Of more importance in our present subject is the theorem that, if the normal velocity be zero at every point of the internal boundary, and if the fluid be at rest at infinity, then </> is everywhere constant. We cannot how- ever infer this at once from the proof of the corresponding theorem in Art. 40. It is true that we may suppose the region limited externally by an infinitely large surface at every point of which d<j>/dn is infinitely small; but it is conceivable that the integral ffd<p/dn . dS, taken over a portion of this surface, might still be finite, in which case the investigation referred to would fail. We proceed therefore as follows. Since the velocity tends to the limit zero at an infinite distance from the internal boundary (S, say), it must be possible to draw a closed surface 2 40-42] Conditions of Determinateness 43 completely enclosing S, beyond which the velocity is everywhere less than a certain value e, which value may, by making 2 large enough, be made as small as we please. Now in any direction from S let us take a point P at such a distance beyond 2 that the solid angle which 2 s.ubtends at it is infinitely small; and with P as centre let us describe two spheres, one just excluding, the other just including S. We shall prove that the mean value of (j> over each of these spheres is, within an infinitely small amount, the same. For if Q, Q' be points of these spheres on a common radius PQQ' , then if Q, Q' fall within 2 the corresponding values of (f> may differ by a finite amount; but since the portion of either spherical surface which falls within X is an infinitely small fraction of the whole, no finite difference in the mean values can arise from this cause. On the other hand, when Q, Q' fall without %, the corresponding values of <£ cannot differ by so much as e . QQ', for e is by definition a superior limit to the rate of variation of <£. Hence, the mean values of <f> over the two spherical surfaces must differ by less than e . QQ' . Since QQ' is finite, whilst e may by taking S large enough be made as small as we please, the difference of the mean values may, by taking P sufficiently distant, be made infinitely small. Now we have seen in Arts. 38, 39 that the mean value of </> over the inner sphere is equal to its value at P, and that the mean value over the outer sphere is (since M = 0) equal to a constant quantity C. Hence, ultimately, the value of <f> at infinity tends everywhere to the constant value C. The same result holds even if the normal velocity be not zero over the internal boundary; for in the theorem of Art. 39 M is divided by r, which is in our case infinite. It follows that if d(f)/dn = at all points of the internal boundary, and if the fluid be at rest at infinity, it must be everywhere at rest. For no lines of motion can begin or end on the internal boundary. Hence such lines, if they existed, must come from an infinite distance, traverse the region occupied by the fluid, and pass off again to infinity; i.e. they must form infinitely long courses between places where <£ has, with an infinitely small amount, the same value (7, which is impossible. The theorem that, if the fluid be at rest at infinity, the motion is deter- minate when the value of — d(f>/dn is given over the internal boundary, follows by the same argument as in Art. 40. Greens Theorem. 42. In treatises on Electrostatics, &c, many important properties of the potential are usually proved by means of a certain theorem due to Green. Of these the most important from our present point of view have already been given; but as the theorem in question leads, amongst other things, to a useful 44 Irrotational Motion [chap, hi expression for the kinetic energy in any case of irrotational motion, some account of it will properly find a place here. Let U, V, W be any three functions which are finite, single-valued and differentiable at all points of a connected region completely bounded by one or more closed surfaces S; let 88 be an element of any one of these surfaces, and I, m, n the direction-cosines of the normals to it drawn inwards. We shall prove in the first place that [l(W + mV+nW)dS=-lj((^+ d ^ (1) where the triple-integral is taken throughout the region, and the double- integral over its boundary. Tf we conceive a series of surfaces drawn so as to divide the region into any number of separate parts, the integral fj(lU + mV + nW)dS, (2) taken over the original boundary, is equal to the sum of the similar integrals each taken over the whole boundary of one of these parts. For, for every element. 8a of a dividing surface, we have, in the integrals corresponding to the parts lying on the two sides of this surface, elements (IU + mV+nW) 8a, and (I'U + m'V + n W) 8a, respectively. But the normals to which I, m, n and V , m ', n refer being drawn inwards in each case, we have V = — I, m! = — m, n' = — n; so that, in forming the sum of the integrals spoken of, the elements due to the dividing surfaces disappear, and we have left only those due to the original boundary of the region. Now let us suppose the dividing surfaces to consist of three systems of planes, drawn at infinitesimal intervals, parallel to yz, zx, xy, respectively. If x, y, z be the co-ordinates of the centre of one of the rectangular spaces thus formed, and 8x, 8y, 8z the lengths of its edges, the part of the integral (2) due to the yz-f&ce nearest the origin is and that due to the opposite face is -(u + i^Sa^SyBz. The sum of these is — dU/dx . 8x8y8z. Calculating in the same way the parts of the integral due to the remaining pairs of faces, we get for the final result Hence (1) simply expresses the fact that the surface-integral (2), taken over the boundary of the region, is equal to the sum of the similar integrals taken 42-43] Green's Theorem 45 over the boundaries of the elementary spaces of which we have supposed it built up. It is evident from (1), or it may be proved directly by transformation of co-ordinates, that if U, V, W be regarded as components of a vector, the expression d_u dv dw dx dy dz is a 'scalar' quantity, i.e. its value is unaffected by any such transformation. It is now usually called the 'divergence' of the vector-field at the point (x, y, z). The interpretation of (1), when (U, V, W) is the velocity of a continuous substance, is obvious. In the particular case of irrotational motion we obtain d ^dS = -ljlv*<j>dxdydz, (3) where 8n denotes an element of the inwardly-directed normal to the surface S. Again, if we put U, V, W = pu, pv, pw, respectively, we reproduce in substance the second investigation of Art. 7. Another useful result is obtained by putting U, V, W = i«/>, v<fi, wcf>, respec- tively, where u, v, w satisfy the relation du dv dw _ dx dy dz throughout the region, and make lu i-mv + nw = over the boundary. We find ii ll!{%+'%+'%)**?-o <*> dy The function cj> is here merely restricted to be finite, single- valued, and con- tinuous, and to have its first differential coefficients finite, throughout the region. 43. Now let 0, (/>' be any two functions which, together with their first and second derivatives, are finite and single-valued throughout the region considered; and let us put d(h' respectively, so that IU + mV + n W =</>•—. Substituting in (1) we find -!J!<tX*<t>'dxdydz ..(5) 46 Irrotational Motion [chap, hi By interchanging </> and <fi we obtain -$JS$V 2 4>dxdydz (6) Equations (5) and (6) together constitute Green's theorem* 44. If <£, (/>' be the velocity-potentials of two distinct modes of irrotational motion of a liquid, so that V 2 </> = 0, V 2 </>' = 0, (1) we obtain jf^dS- jfo'^dS (2) If we recall the physical interpretation of the velocity-potential, given in Art. 18, then, regarding the motion as generated in each case impulsively from rest, we recognize this equation as a particular case of the dynamical theorem that tp r qr =Zpr'qr, where p r} q r and p r ', q r ' are generalized components of impulse and velocity, in any two possible motions of a system f. Again, in Art. 43 (6) let </>' = <\>, and let <f> be the velocity-potential of a liquid. We obtain (^♦gr*«n***~j«« <»> To interpret this we multiply both sides by ^p. Then on the right-hand side — d(j>/dn denotes the normal velocity of the fluid inwards, whilst p<\> is, by Art. 18, the impulsive pressure necessary to generate the motion. It is a proposition in Dynamics J that the work done by an impulse is measured by the product of the impulse into half the sum of the initial and final velocities, resolved in the direction of the impulse, of the point to which it is applied. Hence the right-hand side of (3), when modified as described, expresses the work done by the system of impulsive pressures which, applied to the surface S, would generate the actual motion; whilst the left-hand side gives the kinetic energy of this motion. The formula asserts that these two quantities are equal. Hence if T denote the total kinetic energy of the liquid, we have the very important formula tT — pjj^d8. (4) If in (3), in place of <f), we write d<f)/dx, which will of course satisfy V 2 3$/9#=0, and ipply the resulting theorem to the region included within a spherical surface of radius r * G. Green, Essay on Electricity and Magnetism, Nottingham, 1828, Art. 3 [Mathematical Papers (ed. Ferrers), Cambridge, 1871, p. 3]. f Thomson and Tait, Natural Philosophy, Art. 313, equation (11). X Ibid. Art. 308. 43-45] Kinetic Energy 47 having any point (x, y, z) as centre, then with the same notation as in Art. 39, we have Hence, writing q 2 = u 2 + v 2 + iv 2 , Since this latter expression is essentially positive, the mean value of q 2 , taken over a sphere having any given point as centre, increases with the radius of the sphere. Hence q 2 cannot be a maximum at any point of the fluid, as was proved otherwise in Art. 37. Moreover, recalling the formula for the pressure in any case of irrotational motion of a liquid, viz. &J*+-a-W+F{t\ (6) we infer that, provided the potential Q. of the external forces satisfy the condition V 2 S2=0, (7) the mean value of p over a sphere described with any point in the interior of the fluid as centre will diminish as the radius increases. The place of least pressure will therefore be somewhere on the boundary of the fluid. This has a bearing on the point discussed in Art. 23. 45. In this connection we may note a remarkable theorem discovered by Lord Kelvin*, and afterwards generalized by him into an universal property of dynamical systems started impulsively from rest under prescribed velocity- conditions f. The irrotational motion of a liquid occupying a simply-connected region has less kinetic energy than any other motion consistent with the same normal motion of the boundary. Let The the kinetic energy of the irrotational motion to which the velocity- potential $ refers, and T± that of another motion given by dd> dd> dd> , n \ U = -dx +U< " V = -frj +V °' W = -dI + W °' (8) where, in virtue of the equation of continuity, and the prescribed boundary- condition, we must have duo dvo dwo _ dx dy dz throughout the region, and lu + mv + nw = over the boundary. Further let us write To = ip!f!(u<? + v 2 + w 2 )dxdydz (9) * (W. Thomson) "On the Vis-Viva of a Liquid in Motion," Camb. and Dub. Math. Journ. 1849 [Papers, i. 107]. t Thomson and Tait, Art. 312. 48 Irrotational Motion [chap, hi We find T 1 = T+T - pj'JfUo^ + v ^ + w ^ dxdydz. Since the last integral vanishes, by Art. 42 (4), we have r l\ = T+T Qy (10) which proves the theorem*. 46. We shall require to know, hereafter, the form assumed by the ex- pression (4) for the kinetic energy when the fluid extends to infinity and is at rest there, being limited internally by one or more closed surfaces S. Let us suppose a large closed surface X described so as to enclose the whole of S. The energy of the fluid included between S and % is -*,//♦£*-!,//$.* (ID where the integration in the first term extends over S, that in the second over 2. Since we have, by the equation of continuity, the expression (11) may be written -Ipffa-O^dS-lpfjw-C^dl, (12) where G may be any constant, but is here supposed to be the constant value to which <j) was shewn in Art. 39 to tend at an infinite distance from S. Now the whole region occupied by the fluid may be supposed made up of tubes of flow, each of which must pass either from one point of the internal boundary to another, or from that boundary to infinity. Hence the value of the integral d<j> si dn d1 ' taken over any surface, open or closed, finite or infinite, drawn within the region, must be finite. Hence ultimately, when 2 is taken infinitely large and infinitely distant all round from S, the second term of (12) vanishes, and we have 2T = -pjj(<p-C) d ^dS, (13) where the integration extends over the internal boundary only. If the total flux across the internal boundarv be zero, we have ff d ^dS = 0, on so that (13) may be written 2T = - p jU^dS, (14) simply. * Some extensions of this result are discussed by Leathern, Cambridge Tracts, No. 1, 2nd ed. (1913). They supply further interesting illustrations of Kelvin's general dynamical principle. 45-47] Cyclic Regions 49 On Multiply-connected Regions. 47. Before discussing the properties of irrotational motion in multiply- connected regions we must examine more in detail the nature and classifica- tion of such regions. In the following synopsis of this branch of the geometry of position we recapitulate for the sake of completeness one or two definitions already given. We consider any connected region of space, enclosed by boundaries. A region is 'connected' when it is possible to pass from any one point of it to any other by an infinity of paths, each of which lies wholly in the region. Any two such paths, or any two circuits, which can by continuous variation be made to coincide without ever passing out of the region, are said to be 'mutually reconcileable.' Any circuit which can be contracted to a point without passing out of the region is said to be 'reducible/ Two reconcileable paths, combined, form a reducible circuit. If two paths or two circuits be reconcileable, it must be possible to connect them by a continuous surface, which lies wholly within the region, and of which they form the complete boundary: and conversely. It is further convenient to distinguish between 'simple' and 'multiple' irreducible circuits. A 'multiple' circuit is one which cau by continuous variation be made to appear, in whole or in part, as the repetition of another circuit a certain number of times. A 'simple' circuit is one with which this is not possible. A 'barrier,' or 'diaphragm,' is a surface drawn across the region, and limited by the line or lines in which it meets the boundary. Hence a barrier is necessarily a connected surface, and cannot consist of two or more detached portions. A 'simply-connected' region is one such that all paths joining any two points of it are reconcileable, or such that all circuits drawn within it are reducible. A 'doubly-connected' region is one such that two irreconcileable paths, and no more, can be drawn between any two points A, B of it; viz. any other path joining AB is reconcileable with one of these, or with a combination of the two taken each a certain number of times. In other words, the region is such that one (simple) irreducible circuit can be drawn in it, whilst all other circuits are either reconcileable with this (repeated, if necessary), or are reducible. As an example of a doubly-connected region we may take that enclosed by the surface of an anchor-ring, or that external to such a ring and extending to infinity. Generally, a region such that n irreconcileable paths, and no more, can be drawn between any two points of it, or such that n — 1 (simple) irreducible 50 Irrotational Motion [chap, hi and irreconcileable circuits, and no more, can be drawn in it, is said to be '^-ply-connected.' The shaded portion of the figure on p. 35 is a triply-connected space of two dimensions. It may be shewn that the above definition of an w-ply-connected space is self-consistent. In such simple cases as n = 2, n = 3, this is sufficiently evident without demonstration. 48. Let us suppose, now, that w 7 e have an w-ply-connected region, with n — 1 simple independent irreducible circuits drawn in it. It is possible to draw a barrier meeting any one of these circuits in one point only, and not meeting any of the n — 2 remaining circuits. A barrier drawn in this manner does not destroy the continuity of the region, for the interrupted circuit remains as a path leading round from one side to the other. The order of connection of the region is however diminished by unity; for every circuit drawn in the modified region must be reconcileable with one or more of the n — 2 circuits not met by the barrier. A second barrier, drawn in the same manner, will reduce the order of con- nection again by one, and so on ; so that by drawing n — 1 barriers we can reduce the region to a simply-connected one. A simply-connected region is divided by a barrier into two separate parts; for otherwise it would be possible to pass from a point on one side of the barrier to an adjacent point on the other side by a path lying wholly within the region, which path would in the original region form an irreducible circuit. Hence in an w-ply-connected region it is possible to draw n — 1 barriers, and no more, without destroying the continuity of the region. This property is sometimes adopted as the definition of an ^-ply-connected space. Irrotational Motion in Multiply -connected Spaces. 49. The circulation is the same in any two reconcileable circuits ABC A, A'B'G'A' drawn in a region occupied by fluid moving irrotationally. For the two circuits may be connected by a continuous surface lying wholly within the region; and if we apply the theorem of Art. 32 to this surface, we have, remembering the rule as to the direction of integration round the boundary. I(ABCA) + I(A'C'B'A') = 0, or 1 (ABGA) = I (A'B'C'A'). If a circuit ABGA be reconcileable with two or more circuits A'B'C A' , A"B"G" A" , &c, combined, we can connect all these circuits by a continuous surface which lies wholly within the region, and of which they form the com- plete boundary. Hence J (ABGA) + 1 (A'G'B'A') + / (A"G"B"A") + &c. = 0, or I{ABGA) = I{A'B , C'A , ) + I(A"B"G"A") + &z.; 47-50] Cyclic Velocity -Potentials 51 i.e. the circulation in any circuit is equal to the sum of the circulations in the several members of any set of circuits with which it is reconcileable. Let the order of connection of the region be n + 1, so that n independent simple irreducible circuits a 1} a 2 , ... a n can be drawn in it; and let the circu- lations in these be k 1} k 2 , ... fc n , respectively. The sign of any k will of course depend on the direction of integration round the corresponding circuit ; let the direction in which k is estimated be called the positive direction in the circuit. The value of the circulation in any other circuit can now be found at once. For the given circuit is necessarily reconcileable with some com- bination of the circuits a lt a 2 , ... a n ; say with a x taken p ± times, a 2 taken p 2 times and so on, where of course any p is negative when the corre- sponding circuit is taken in the negative direction. The required circulation then is PifC!-{-p 2 /c 2 + ...+p n K n (1) Since any two paths joining two points A, B of the region together form a circuit, it follows that the values of the flow in the two paths differ by a quantity of the form (1), where, of course, in particular cases some or all of the p's may be zero. 50. Let us denote by — <p the flow to a variable point P from a fixed point A, viz. (/> = — (udcc + vdy + wdz) (2) J A So long as the path of integration from A to P is not specified, </> is indeter- minate to the extent of a quantity of the form (1). If however n barriers be drawn in the manner explained in Art. 48, so as to reduce the region to a simply-connected one, and if the path of integration in (2) be restricted to lie within the region as thus modified {i.e. it is not to cross any of the barriers), then <f> becomes a single-valued function, as in Art. 35. It is continuous throughout the modified region, but its values at two adjacent points on opposite sides of a barrier differ by ± k. To derive the value of (j) when the integration is taken along any path in the unmodified region we must subtract the quantity (1), where any p denotes the number of times this path crosses the corresponding barrier. A crossing in the positive direction of the circuits interrupted by the barrier is here counted as positive, a crossing in the opposite direction as negative. By displacing P through an infinitely short space parallel to each co-ordinate axis in succession, we find dd> dd> deb U ' V > W = ~£' ~Jy- ~Tz> so that </> satisfies the definition of a velocity- potential (Art. 17). It is now however a many-valued or cyclic function ; i.e. it is not possible to assign to every point of the original region a unique and definite value of <£, such values /: 52 Irrotational Motion [chap, in forming a continuous system. On the contrary, whenever P describes an irre- ducible circuit, </> will not, in general, return to its original value, but will differ from it by a quantity of the form (1). The quantities *i, k 2 , ... /c n , which specify the amounts by which <£ decreases as P describes the several independent circuits of the region, may be called the 'cyclic constants' of </>. It is an immediate consequence of the 'circulation-theorem' of Art. 33 that under the conditions there presupposed the cyclic constants do not alter with the time. The necessity for these conditions is exemplified in the problem of Art. 29, where the potential of the extraneous forces is itself a cyclic function. The foregoing theory may be illustrated by the case of Art. 27 (2), where the region (as limited by the exclusion of the origin, since the formula would give an infinite velocity there) is doubly-connected ; for we can connect any two points A, B of it by two irreconcileable paths passing on opposite sides of the axis of 2, e.g. ACB, ADB in the figure. The portion of the plane zx for which x is positive, may be taken as a barrier, and the region is thus made simply-connected. The circulation in any circuit meeting this barrier once only, e.g. in ACB DA, is •2tt fijr. rd&, or 27r/x. That in any circuit not meeting the barrier is zero. In the modified region <£ may be put equal to a single- valued function, viz. — fid, but its value on the positive side of the barrier is zero, that at an adjacent point on the negative is — 2-rrfi. More complex illustrations of irrotational motion in multiply-connected spaces of two dimensions will present themselves in the next chapter. 51. Before proceeding further we may briefly indicate a somewhat different method of presenting the above theory. Starting from the existence of a velocity-potential as the characteristic of the class of motions which we propose to study, and adopting the second definition of an n+ 1 -ply-connected region, indicated in Art. 48, we remark that in a simply-connected region every equipotential surface must either be a closed surface, or else form a barrier dividing the region into two separate parts. Hence, supposing the whole system of such surfaces drawn, we see that if a closed curve cross any given equipotential surface once it must cross it again, and in the opposite direction. Hence, corresponding to any element of the curve, included between two consecutive equipotential surfaces, we have a second element such that the flow along it, being equal to the difference between the corresponding values of </>, is equal and opposite to that along the former; so that the circulation in the whole circuit is zero. If however the region be multiply-connected, an equipotential surface may form a barrier without dividing it into two separate parts. Let as many such surfaces be drawn as is possible without destroying the con- tinuity of the region. The number of these cannot, by definition, be greater 50-52] Multiple Connectivity 53 than n. Every other equipotential surface which is not closed will be re- concileable (in an obvious sense) with one or more of these barriers. A curve drawn from one side of a barrier round to the other, without meeting any of the remaining barriers, will cross every equipotential surface reconcileable with the first barrier an odd number of times, and every other equipotential surface an even number of times. Hence the circulation in the circuit thus formed will not vanish, and </> will be a cyclic function. In the method adopted above we have based the whole theory on the equations dw dv _ n du dw _ d_v _ du _ , . dy dz ' dz dx ' dx dy and have deduced the existence and properties of the velocity-potential in the various cases as necessary consequences of these. In fact, Arts. 34. 35, and 49, 50 may be regarded as an inquiry into the nature of the solution of this system of differential equations, as depending on the character of the region through which they hold. The integration of (3), when we have, on the right-hand side, instead of zero, known functions of x, y, z, will be treated in Chapter vn. 52. Proceeding now, as in Art. 36, to the particular case of an incom- pressible fluid, we remark that whether <f> be cyclic or not, its first derivatives d(f>/dx, d(j>/dy, d4>/dz, and therefore all the higher derivatives, are essentially single-valued functions, so that <j> will still satisfy the equation of continuity V 2 = O, (1) or the equivalent form II ~ dS — 0, (2) where the surface-integration extends over the whole boundary of any portion of the fluid. The theorem (a) of Art. 40, viz. that <£ must be constant throughout the interior of any region at every point of which (1) is satisfied, if it be constant over the boundary, still holds when the region is multiply-connected. For <£, being constant over the boundary, is necessarily single-valued. The remaining theorems of Art. 40, being based on the assumption that the stream-lines cannot form closed curves, will require modification. We must introduce the additional condition that the circulation is to be zero in each circuit of the region. Removing this restriction, we have the theorem that the irrotational motion of a liquid occupying an n-ply-connected region is determinate when the normal velocity at every point of the boundary is prescribed, as well as the value of the circulation in each of the n independent and irreducible circuits which can be drawn in the region. For if <j> 1} <£ 2 be the (cyclic) velocity-potentials of two motions satisfying the above conditions, then 54 Irrotational Motion [chap, hi <£ = <£i — $2 is a single- valued function which satisfies (1) at every point of the region, and makes d<f)/dn = at every point of the boundary. Hence, by Art. 40, cf) is constant, and the motions determined by </>i and </> 2 are identical. The theory of multiple connectivity seems to have been first developed by Riemann* for spaces of two dimensions, a propos of his researches on the theory of functions of a complex variable, in which connection also cyclic functions satisfying the equations da 2 dy' 2 through multiply-connected regions, present themselves. The bearing of the theory on Hydrodynamics and the existence in certain cases of many-valued velocity-potentials were first pointed out by von Helmholtzt. The subject of cyclic irrotational motion in multiply-connected regions was afterwards taken up and fully investigated by Lord Kelvin in the paper on vortex-motion already referred to J. Kelvins Extension of Greens Theorem. 53. It was assumed in the proof of Green's theorem that </> and <f>' were both single-valued functions. If either be a cyclic function, as may be the case when the region to which the integrations in Art. 43 refer is multiply- connected, the statement of the theorem must be modified. Let us suppose, for instance, that cj> is cyclic; the surface-integral on the left-hand side of Art. 43 (5), and the second volume-integral on the right-hand side, are then indeterminate, on account of the indeterminateness in the value of <£ itself. To remove this indeterminateness, let the barriers necessary to reduce the region to a simply-connected one be drawn, as explained in Art. 48. We may now suppose <fi to be continuous and single-valued throughout the region thus modified ; and the equation referred to will then hold, provided the two sides of each barrier be reckoned as part of the boundary of the region, and therefore included in the surface-integral on the left-hand side. Let oVi, be an element of one of the barriers, kx the cyclic constant corresponding to that barrier, d<f>'/dn the rate of variation of <£' in the positive direction of the normal to 8<ri'. Since, in the parts of the surface-integral due to the two sides of So"!, d<j>'/dn is to be taken with opposite signs, whilst the value of </> on the positive side exceeds that on the negative side by tci, we get finally for the element of the integral due to So"i, the value /c^fi/dn. Bai. Hence Art. 43 (5) becomes, in the altered circumstances, * Grundlagen fiir eine allgemeine Theorie der Funetionen einer veranderlichen complexen Gr'osse, Gottingen, 1851 [Matheviatische Werke, Leipzig, 1876, p. 3]. Also: "Lehrsatze aus der Analysis Situs," Crelle, liv. (1857) [Werke, p. 84]. t Crelle, lv. (1858). J See also Kirchhoff , ' ' Ueber die Krai te welche zwei unendlich diinne starre Hinge in einer Fliissigkeit scheinbar auf einander ausiiben konnen," Crelle, lxxi. (1869) [Gesammelte Abhand- lungen, Leipzig, 1882, p. 404]. 52-54] Extension of Green's Theorem 55 where the surface-integrations indicated on the left-hand side extend, the first over the original boundary of the region only, and the rest over the several barriers. The coefficient of any k is evidently minus the total flux across the corresponding barrier, in a motion of which <f>' is the velocity- potential. The values of <j> in the first and last terms of the equation are to be assigned in the manner indicated in Art. 50. If 4>' also be a cyclic function, having the cyclic constants «/, k 2 , &c., then Art. 43 (6) becomes in the same way h't^A\t^A\l da 2 + • . . Equations (1) and (2) together constitute Lord Kelvin's extension of Green's theorem. 54. If <f), ()>' are both velocity-potentials of a liquid, we have V 2 (/> = 0, V 2 f = 0, (3) and therefore <£ ■— dS + K\ \\-^- do-} + k 2 -^- da 2 + . . . -JJ'S^JS^Wf^ (4 > To obtain a physical interpretation of this theorem it is necessary to explain in the first place a method, imagined by Lord Kelvin, of generating any given cyclic irrotational motion of a liquid "in a multiply-connected space. Let us suppose the fluid to be enclosed in a perfectly smooth and flexible membrane occupying the position of the boundary. Further, let n barriers be drawn, as in Art. 48, so as to convert the region into a simply-connected one, and let their places be occupied by similar membranes, infinitely thin, and destitute of inertia. The fluid being initially at rest, let each element of the first-mentioned membrane be suddenly moved inwards with the given (positive or negative) normal velocity — d(f>/dn, whilst uniform impulsive pressures Kip,/c 2 p,... K n p are simultaneously applied to the negative sides of the respective barrier-membranes. The motion generated will be characterized by the following properties. It will be irrotational, being generated from rest ; the normal velocity at every point of the original boundary will have the prescribed value ; the values of the impulsive pressure at two adjacent points on opposite sides of a membrane will differ by the corresponding value of Kp, and the values of the velocity-potential will therefore differ by the corresponding value of k ; finally, the motion on one side of a barrier will be continuous with that on the other. To prove the last statement we remark, first, that the velocities normal to the barrier at two adjacent points on 56 Irrotational Motion [chap, hi opposite sides of it are the same, being each equal to the normal velocity of the adjacent portion of the membrane. Again, if P, Q be two consecutive points on a barrier, and if the corresponding values of <£ be on the positive side </> P , (pQ, and on the negative side $ V P , <£ v q, we have and therefore (f> Q — cj> P — $ V Q — <£ V P , i.e., if PQ = Bs, d(f>/ds = dfi/ds. Hence the tangential velocities at two adjacent points on opposite sides of the barrier also agree. If then we suppose the barrier-membranes to be liquefied immediately after the impulse, we obtain the irrotational motion in question. The physical interpretation of (4), when multiplied by — p, now follows as in Art. 44. The values of p/c are additional components of momentum, and those of —ffd<f>/dn.d(r, the fluxes through the various apertures of the region, are the corresponding generalized velocities. 55. If in (2) we put (f> r = <j>, and suppose $ to be the velocity-potential of an incompressible fluid, we find = - p jj'i >d £ ds - p ' (i \\ d £ d ' 7i - pK *\j d £ d ' T *- (5) The last member of this formula has a simple interpretation in terms of the artificial method of generating cyclic irrotational motion just explained. The first term has already been recognized as equal to twice the work done by the impulsive pressure pcf> applied to every part of the original boundary of the fluid. Again, pK X is the impulsive pressure applied, in the positive direction, to the infinitely thin massless membrane by which the place of the first barrier was supposed to be occupied ; so that the expression -»/j d< t> ^ denotes the work done by the impulsive forces applied to that membrane ; and so on. Hence (5) expresses the fact that the energy of the motion is equal to the work done by the whole system of impulsive forces by which we may suppose it generated. In applying (5) to the case where the fluid extends to infinity and is at rest there, we may replace the first term of the third member by -pjj(4>-C)QdS, (6) where the integration extends over the internal boundary only. The proof 54-56] Kinetic Energy 57 is the same as in Art 46. When the total flux across this boundary is zero, this reduces to P JK> ^ The minimum theorem of Lord Kelvin, given in Art. 45, may now be extended as follows: The irrotational motion of a liquid in a multiply-connected region has less kinetic energy than any other motion consistent with the same normal motion of the boundary and the same value of the total flux through each of the several independent channels of the region. The proof is left to the reader. Sources and Sinks. 56. The analogy with the theories of Electrostatics, the Steady Flow of Heat, &c, may be carried further by means of the conception of sources and sinks. A ' simple source ' is a point from which fluid is imagined to flow out uniformly in all directions. If the total flux outwards across a small closed surface surrounding the point be m, then m is called the ' strength ' of the source. A negative source is called a 'sink.' The continued existence of a source or a sink would postulate of course a continual creation or annihila- tion of fluid at the point in question. The velocity-potential at any point P, due to a simple source, in a liquid at rest at infinity, is (f> = m/Awr, (1) where r denotes the distance of P from the source. For this gives a radial flow from the point, and if &S, = r 2 ^, be an element of a spherical surface having its centre at the source, we have -w d ^dS = m, or a constant, so that the equation of continuity is satisfied, and the flux outwards has the value appropriate to the strength of the source. A combination of two equal and opposite sources ± w', at a distance 8s apart, where, in the limit, Bs is taken to be infinitely small, and m' infinitely great, but so that the product m'Bs is finite and equal to yu, (say), is called a ' double source ' of strength /m, and the line Ss, considered as drawn in the direction from — m' to 4- m' , is called its axis. To find the velocity-potential at any point (a, y, z) due to a double source ~"L[ 58 Irrotational Motion [chap, hi fju situate at (#', y', z'), and having its axis in the direction (I, m, n), we remark that, / being any continuous function, /O' + IBs, y' + mBs, z' + nSs) -f{x\ y', z') ultimately. Hence, putting/ (a/, y' , z') — m' \kirr, where r ={{ x -oc'f + {y-y'? + (z-z'y}h wefind ^=H l i + m w +n i)^ (2) i<L + rn <L+ n <L)l (3) dx dy dzj r 1 _ n cosS" ... -^T~^~' W where, in the latter form, ^ denotes the angle which the line r, considered as drawn from (V, y', z') to (%, y, z), makes with the axis (I, m, n). We might proceed, in a similar manner (see Art. 82), to build up sources of higher degrees of complexity, but the above is sufficient for our immediate purpose. Finally, we may imagine simple or double sources, instead of existing at isolated points, to be distributed continuously over lines, surfaces, or volumes. 57. We can now prove that any continuous acyclic irrotational motion of a liquid mass may be regarded as due to a distribution of simple and double sources over the boundary. This depends on the theorem, proved in Art. 44, that if <£, <£' be any two single-valued functions which satisfy V 2 <£ = 0, V 2 0' = O throughout a given region, then IKt'^MK^ (5) where the integration extends over the whole boundary. In the present application, we take cj) to be the velocity-potential of the motion in question, and put <f> = 1/r, the reciprocal of the distance of any point of the fluid from a fixed point P. We will first suppose that P is in the space occupied by the fluid. Since </>' then becomes infinite at P, it is necessary to exclude this point from the region to which the formula (5) applies; this may be done by describing a small spherical surface about P as centre. If we now suppose S2 to refer to this surface, and &S to the original boundary, the formula gives 56-58] Sources and Sinks 59 At the surface 2 we have d/dn (1/r) = — 1/r 2 ; hence if we put 81=r 2 d&, and finally make r-*-0, the first integral on the left-hand becomes = — 4<7r</>p, where (f> P denotes the value of (f> at P, whilst the first integral on the right vanishes. Hence *--BjJfif«+eK®« « This gives the value of <£ at any point P of the fluid in terms of the values of <j> and d<f>/dn at the boundary. Comparing with the formulae (1) and (2) we see that the first term is the velocity-potential due to a surface distribution of simple sources, with a density — d<f>/dn per unit area, whilst the second term is the velocity-potential of a distribution of double sources, with axes normal to the surface, the density being <j>. It will appear from equation (10), below, that this is only one out of an infinite number of surface-distributions which will give the same value of throughout the interior. When the fluid extends to infinity in every direction and is at rest there, the surface-integrals in (7) may, on a certain understanding, be taken to refer to the internal boundary alone. To see this, we may take as external boundary an infinite sphere having the point P as centre. The corresponding part of the first integral in (7) vanishes, whilst that of the second is equal to G, the constant value to which, as we have seen in Art. 41, <£ tends at infinity. It is convenient, for facility of statement, to suppose (7 = 0; this is legitimate since we may always add an arbitrary constant to <£. When the point P is external to the surface, <f>' is finite throughout the original region, and the formula (5) gives at once *~cJ£&*+cKG)« < 8) where, again, in the case of a liquid extending to infinity, and at rest there, the terms due to the infinitely distant part of the boundary may be omitted. 58. The distribution expressed by (7) can, further, be replaced by one of simple sources only, or of double sources only, over the boundary. Let </> be the velocity-potential of the fluid occupying a certain region, and let (/>' now denote the velocity- potential of any possible acyclic irrotational motion through the rest of infinite space, with the condition that </>, or <£', as the case may be, vanishes at infinity. Then, if the point P be internal to the first region, and therefore external to the second, we have °--ism^M*M)^\ where Bn, 8n' denote elements of the normal to dS, drawn inwards to the 60 Irrotational Motion [chap, hi first and second regions respectively, so that d/dn' = — d/dn. By addition, we have ♦.--sjj?e+&)«+sjj<*-*4©« ■■■» The function <f>' will be determined by the surface-values of <£' or d<j>/dn', which are as yet at our disposal. Let us in the first place make <f> = <j> at the surface. The tangential velocities on the two sides of the boundary are then continuous, but the normal velocities are discontinuous. To assist the ideas, we may imagine a liquid to fill infinite space, and to be divided into two portions by an infinitely thin vacuous sheet within which an impulsive pressure pcf> is applied, so as to generate the given motion from rest. The last term of (10) disappears, so that *>»*$&*%)"■ <"> that is, the motion (on either side) is that due to a surface-distribution of simple sources, of density Secondly, we may suppose that d<j>'/dn = d(f>/dn over the boundary. This gives continuous normal velocity, but discontinuous tangential velocity, over the original boundary. The motion may in this case be imagined to be generated by giving the prescribed normal velocity — d<\>jdn to every point of an infinitely thin membrane coincident in position with the boundary. The first term of (10) now vanishes, and we have fc-sjfa-^s©** < 12 > shewing that the motion on either side may be conceived as due to a surface- distribution of double sources, with density It may be shewn that the above representations of (f> in terms of simple sources alone, or of double sources alone, are unique; whereas the representa- tion of Art. 57 is indeterminate f. It is obvious that cyclic irrotational motion of a liquid cannot be reproduced by any arrangement of simple sources. It is easily seen, however, that it may be represented by a certain distribution of double sources over the boundary, together with a uniform distri- bution of double sources over each of the barriers necessary to render the region occupied by the fluid simply-connected. In fact, with the same notation as in Art. 53, we find * This investigation was first given by Green, from the point of view of Electrostatics, I.e. ante p. 46. f Cf. Larmor, "On the Mathematical Expression of the Principle of Huyghens," Proc. Lond. Math. Soc. (2) i. 1 (1903) [Math, and Phys. Papers, Cambridge, 1929, ii. 240]. 58] Surface-Distributions 61 where <p is the single-valued velocity-potential which obtains in the modified region, and <f>' is the velocity-potential of the acyclic motion which is generated in the external space when the proper normal velocity —dcfy/dn is given to each element 8S of a membrane coincident in position with the original boundary. Another mode of representing the irrotational motion of a liquid, whether cyclic or not, will present itself in the chapter on Vortex Motion. We here close this account of the theory of irrotational motion. The mathematical reader will doubtless have noticed the absence of some im- portant links in the chain of our propositions. For example, apart from physical considerations, no proof has been offered that a function <f> exists which satisfies the conditions of Art. 36 throughout any given simply- connected region, and has arbitrarily prescribed values over the boundary. The formal proof of 'existence-theorems' of this kind is not attempted in the present treatise. For a review of the literature of this part of the subject the reader may consult the authors cited below*. * H. Burkhardt and W. F. Meyer, "Potentialtheorie," and A. Sommerfeld, "Randwerth- aufgaben in der Theorie d. part. Diff.-Gleichungen," Encyc. d. math. Wiss. ii. (1900). CHAPTEE IV MOTION OF A LIQUID IN TWO DIMENSIONS 59. If the velocities u, v be functions of x, y only, while w is zero, the motion takes place in a series of planes parallel to xy, and is the same in each of these planes. The investigation of the motion of a liquid under these circumstances is characterized by certain analytical peculiarities; and the solutions of several problems of great interest are readily obtained. Since the whole motion is known when we know that in the plane z = 0, we may confine our attention to that plane. When we speak of points and lines drawn in it, we shall understand them to represent respectively the straight lines parallel to the axis of z, and the cylindrical surfaces having their generating lines parallel to the axis of z> of which they are the traces. By the flux across any curve we shall understand the volume of fluid which in unit time crosses that portion of the cylindrical surface, having the' curve as base, which is included between the planes z = 0, z — 1. Let i, P be any two points in the plane xy. The flux across any two lines joining AP is the same, provided they can be reconciled without passing out of the region occupied by the moving liquid; for otherwise the space included between these two lines would be gaining or losing matter. Hence if A be fixed, and P variable, the flux across any line AP is a function of the position of P. Let yfr be this function; more precisely, let yjr denote the flux across A P from right to left, as regards an observer placed on the curve, and looking along it from A in the direction of P. Analytically, if I, m be the direction-cosines of the normal (drawn to the left) to any element 8s of the curve, we have yfr= I (lu + mv)ds (1) If the region occupied by the liquid be aperiphractic (see p. 40), i/r is neces- sarily a single-valued function, but in periphractic regions the value of -v/r may depend on the nature of the path A P. For spaces of two dimensions, however, periphraxy and multiple-connectivity become the same thing, so that the properties of i/r, when it is a many-valued function, in relation to the nature of the region occupied by the moving liquid, may be inferred from Art. 50, where we have discussed the same question with regard to </>. The cyclic constants of yjr, when the region is periphractic, are the values of the flux across the closed curves forming the several parts of the internal boundary. 59-eo] Stream- Function 63 A change, say from A to B, of the point from which yjr is reckoned has merely the effect of adding a constant, viz. the flux across a line BA, to the value of yfr; so that we may, if we please, regard ijr as indeterminate to the extent of an additive constant. If P move about in such a manner that the value of yjr does not alter, it will trace out a curve such that no fluid anywhere crosses it, i.e. a stream-line. Hence the curves yjr = const, are the stream-lines, and -\jr is called the 'stream- function.' If P receive an infinitesimal displacement PQ (= By) parallel to y, the increment of yfr is the flux across PQ from right to left, i.e. 8yjr = — u. PQ, or «-- 1 ^ Again, displacing P parallel to x, we find in the same way -3 < 3 > The existence of a function \jr related to u and v in this manner might also have been inferred from the form which the equation of continuity takes in this case, viz. du dv n " 3 -* + ar 0> "; ; (4) which is the analytical condition that udy — vdx should be an exact differential *. The foregoing considerations apply whether the motion be rotational or irrotational. The formulae for the components of vorticity, given in Art. 30, become *-* <-* *-£'■-■?£ < 5 > so that in irrotational motion we have dx 2 + dy 2 w 60. In what follows we confine ourselves to the case of irrotational motion, which is, as we have already seen, characterized by the existence, in addition, of a velocity-potential <£, connected with u, v by the relations d<j> dcf> U = ~dx> V = ~ d y> W and, since we are considering the motion of incompressible fluids only, satisfying the equation of continuity dx 2 ' dy' +H-° ( 2 ) * The function \j/ was introduced in this way by Lagrange, Noav. mem. de VAcad. de Berlin, 1781 [Oeuvres, iv. 720]. The kinematical interpretation is due to Eankine, "On Plane Water- Lines in Two Dimensions," Phil. Trails. 1864 [Miscellaneous Scientific Papers, London, 1881, p.,495]. 64 Motion of a Liquid in Two Dimensions [chap, iv The theory of the function (f>, and the relation between its properties and the nature of the two-dimensional space through which the irrotational motion holds, may be readily inferred from the corresponding theorems in three dimensions proved in the last chapter. The alterations, whether of enunciation or of proof, which are requisite to adapt these to the case of two dimensions are for the most part purely verbal. For instance, we have the theorem that the mean value of <f> over the circumference of a circle is equal to its value at the centre, provided the circle can be contracted to a point, remaining always within the region occupied by the fluid. Again, if this region extends to infinity, being bounded internally by one or more closed curves, and if the velocities tend to a zero limit at infinity, the value of <f> tends there to a constant limit, provided the total flux across the internal boundaries is zero. This latter proviso is now essential. The fundamental solution of the equation (2) has the form <£ = G log r, where r denotes distance from a fixed point. This is the case of a two-dimen- sional source, for if we write * — £logr (3) the flux outwards across a circle surrounding the point is -^. 27rr = ra (4) The constant m accordingly measures the output, or ' strength ', of the source. We get essentially the same result if we imagine point sources of the type explained in Art. 56 to be distributed with uniform line-density m along its axis of z. The velocity in that case will be in the direction of r, and equal to ra/27rr, consistently with (3). We have here the conception of a 'line-source' (in three dimensions). For a double source, or 'doublet', as it is sometimes called, we have the formula *-ll^r) (5) where the symbol d/ds indicates a space-differentiation in the direction of the axis of the source. If ^ be the angle which direction of r increasing makes with this axis, we have Br = — Bs cos ^, and therefore * = £^ W Again we might establish a system of formulae analogous to those of Art. 58. In particular, corresponding to Art. 58 (12), we have fc— i/(* -*') J; ( lo s *•><**• < 7 > 60-60 a] Electrical Analogies 65 giving the value of (/> in any region in terms of a distribution of double sources over the boundary. This will apply to the case of a fluid unlimited externally, provided the velocities tend to zero at infinity, and that the total flux outwards is zero. As in Art. 58 the function </>' refers to the space within the inner boundary, and is subject to the condition that d<f>'/dn = d<j>/dn at this boundary. A deduction from this formula will be given presently (Art. 72 a). 60 a. The foregoing kinematical relations have exact analogies in the theory of electric conduction. In the case of a uniform plane sheet we have . dv dv m ^-~ 5P ag = -W () with ¥+¥ = <>> W dec oy where (/, g) is the current density, V is the electric potential, and a is the specific resistance of the material. If we write u = af, v = ag t <f>=V, (3) these become identical with the hydrodynamical relations. This has suggested a practical method of solution of two-dimensional hydrokinetic problems. The current sheet may consist of a thin layer of feebly conducting fluid (H 2 S0 4 ) contained in a rectangular tank, two opposite walls of which are metallic and maintained at a constant difference of potential whilst the remaining walls (and the bottom) are insulators. The equipotential lines, to which the current lines are orthogonal, are easily traced electrically, and in this way practical solutions can be obtained of problems of flow of a stream past an obstacle (represented by a non-conducting disk in the electrical experiment) which are not easily treated by analysis*. Again, instead of (3) we may put u = — ag, v = crf, -\jr = — V. (4) The hydrodynamical relations are satisfied, but the stream- lines are now represented by the lines of equal electric potential, and can therefore be found directly. An obstacle has now to be represented by a disk whose conductivity so greatly exceeds that of the surrounding stratum that it may be regarded as practically perfect. This analogy has the further advantage that circulation can also be represented. For if (I, m) be the direction of the outward normal to the contour of the obstacle, the circulation is j(lv — mu)ds= a j(lf+ mg)ds, (5) * For experimental details reference should be made to E. F. Relf , Phil. Mag. (6) xlviii. (1924). As a test of the method the diagram on p. 86 infra was reproduced with remarkable accuracy. The circulation round a lamina was also determined and compared with theory. 66 Motion of a Liquid in Two Dimensions [chap, iv and is therefore proportional to the total current outwards in the electric analogy. For this purpose the disk is connected with one terminal of a suitable battery, the other terminal being connected with one of the conducting walls of the tank. 61. The kinetic energy I 1 of a portion of fluid bounded by a cylindrical surface whose generating lines are parallel to the axis of z, and by two planes perpendicular to the axis of z at unit distance apart, is given by the formula »-'/Ji©" + (gn**~' \*i* « where the surface-integral is taken over the portion of the plane xy cut off by the cylindrical surface, and the line-integral round the boundary of this portion. Since d<fi/dn =■■ — df/ds, the formula (1) may be written 2T = pf<t>d+, (2) the integration being carried in the positive direction round the boundary. If we attempt by a process similar to that of Art. 46 to calculate the energy in the case where the region extends to infinity, we find that its value is infinite, except when the total flux outwards (M) is zero. For if we introduce a circle of great radius r as the external boundary of the portion of the plane xy considered, we find that the corresponding part of the integral on the right-hand side of (1) increases indefinitely with r. The only excep- tion is when M=0, in which case we may suppose the line-integral in (1) to extend over the internal boundary only. If the cylindrical part of the boundary consist of two or more separate portions one of which embraces all the rest, the enclosed region is multiply- connected, and the equation (1) needs a correction, which may be applied exactly as in Art. 55. * Conformal Transf of motions. 62. The functions <f> and f are connected by the relations d± = Hi ^ = _?^ (if-* dx dy ' dy dx ' '" ** These conditions are fulfilled by equating <f> + if, where i stands as usual for \/( — 1 ), to any ordinary algebraic or transcendental function of x + iy, say </> + *> =/(* + «» (2) For then g- (0 4- if) = if (x + iy) = i — (<f> + if), (3) whence, equating separately the real and the imaginary parts, we see that the equations (1) are satisfied. 60a-62] Complex Variable 67 Hence any assumption of the form (2) gives a possible case of irrotational motion. The curves (j> = const, are the curves of equal velocity-potential, and the curves yjr = const, are the stream -lines. Since, by (1), d(j> d-yjr dcf) dyjr _ dx dx dy dy we see that these two systems of curves cut one another at right angles, as already proved. Since the relations (1) are unaltered when we write — -v/r for (f>, and cf> for yfr, we may, if we choose, look upon the curves ty = const, as the equipotential curves, and the curves <£ = const, as the stream-lines; so that every assumption of the kind indicated gives us two possible cases of irrotational motion. For shortness, we shall through the rest of this chapter follow the usual notation of the Theory of Functions, and write z = x + iy, (4) W = $ 4- iyfr (5) From a modern point of view, the fundamental property of a function of a complex variable is that it has a definite differential coefficient with respect to that variable*. If cf), ijr denote any functions whatever of x and y, then corresponding to every value of x + iy there must be one or more definite values of <£ + iyjr; but the ratio of the differential of this function to that of x + iy, viz. 8+ + JS+ or @ + *to) fe+ C| + ^) Sy &x+ i8y ' Sso + i8y depends in general on the ratio hx : hy. The condition that it should be the same for all values of the latter ratio is d A +i w =i m +i m (6) dy dy \dx dx J ' which is equivalent to (1) above. This property was adopted by Riemann as the definition of a function of the complex variable x 4- iy; viz. such a function must have, for every assigned value of the variable, not only a definite value or system of values, but also for each of these values a definite differential coefficient. The advantage of this definition is that it is quite independent of the existence of an analytical expression for the function. If the complex quantities z and w be represented geometrically after the manner of Argand and Gauss, the differential coefficient dwjdz may be interpreted as the operator which transforms an infinitesimal vector Bz into the corresponding vector Sw. It follows then, from the above property, that corresponding figures in the planes of z and w are similar in their infinitely small parts. * See, for example, Forsyth, Theory of Functions, 3rd ed., Cambridge, 1918, cc. i., ii. 68 Motion of a Liquid in Two Dimensions [chap, iv For instance, in the plane of w the straight lines <j> = const., yjr = const., where the constants have assigned to them a series of values in arithmetical progression, the common difference being infinitesimal and the same in each case, form two systems of straight lines at right angles, dividing the plane into infinitely small squares. Hence in the plane xy the corresponding curves <f> = const., yjr = const., the values of the constants being assigned as before, cut one another at right angles (as has already been proved otherwise) and divide the plane into infinitely small squares. Conversely, if 0, yjs be any two functions of x, y such that the curves = me, \jr = ne, where e is infinitesimal, and m, n are any integers, divide the plane xy into elementary squares, it is evident geometrically that dx _ dy dx _ _dy If we take the upper signs, these are the conditions that x + iy should be a function of (j> + iyfr. The case of the lower signs is reduced to this by reversing the sign of yj/. Hence the equation (2) contains the complete solution of the problem of conformal representation of one plane on another*. The similarity of corresponding infinitely small portions of the planes w and z breaks down at points where the differential coefficient dw/dz is zero or infinite. Since d ™ = f + i d -±, (7)* dz ox Ox the corresponding value of the velocity, in the hydrodynamical application, is zero or infinite. In all physical applications, w must be a singl^yaljied, or at mos^L-cyclic function of z in the sense of Art. 50, throughout the region with which we are concerned. Hence in the case of a 'multiform' function, this region must be confined to a single sheet of the corresponding Riemann's surface, and 'branch-points' therefore must not occur in its interior. 63. We can now proceed to some applications of the foregoing method. First let us assume w = A z n , A being real. Introducing polar co-ordinates, r, 0, we have (j> = Ar n cos nO, y\r = Ar n sin nO. [ The following cases may be noticed. 1°. If n « 1, the stream-lines are a system of straight lines parallel to x, and the equipotential curves are a similar system parallel to y. In this case any corresponding figures in the planes of w and z are similar, whether they be finite or infinitesimal. * Lagrange, " Sur la construction des cartes geographiques," Nouv. mem. de VAcad. de Berlin, 1779 [Oeuvres, iv. 636]. For the further history of the problem, see Forsyth, Theory of Functions, c. xix. 62-64] Examples 69 2°. If n = 2, the curves <f> = const, are a system of rectangular hyperbolas having the axes of co-ordinates as their principal axes, and the curves yfr = const, are a similar system, having the co-ordinate axes as asymptotes. The lines 6 = 0, 6 = ^ir are parts of the same stream-line yfr = 0, so that we may take the positive parts of the axes of x, y as fixed boundaries, and thus obtain the case of a fluid in motion in the angle between two perpendicular walls. 3°. If n = — 1, we get two systems of circles touching the axes of co-ordinates at the origin. Since now <£ = A/r . cos 6, the velocity at the origin is infinite; we must therefore suppose the region to which our formulae apply to be limited internally by a closed curve. 4°. If n = — 2, each system of curves is composed of a double system of lemniscates. The axes of the system <f> = const, coincide with x or y; those of the system -*|r = const, bisect the angles between these axes. 5°. By properly choosing the value of n we get a case of irrotational motion in which the boundary is composed of two rigid walls inclined at any angle a. The equation of the stream-lines being r 11 sin nO = const., (2) we see that the lines 6 = 0, 6 = irjn are parts of the same stream-line. Hence if we put n = irja, we obtain the required solution in the form (/> = Ar a cos — > yfr = Ar a sin — {?) The component velocities along and perpendicular to r are — A-r a cos~, and A~r a sin— > \V and are therefore zero, finite, or infinite at the origin, according as a is less than, equal to, or greater than ir. 64. We take some examples of cyclic functions. 1°. The assumption w = — /j,\ogz, (1) where /j, is real, gives <f> = — /jl log r, ^ = — fi6 (2) The velocity at a distance r from the origin is fi/r; this point must therefore be isolated by drawing a closed curve round it. If we take the radii 6 = const, as the stream-lines we get the case of a (two-dimensional) source, of strength lir^ at the origin. (See Art. 60.) If the circles r = const, be taken as stream-lines we have the case of Art. 27; the motion is now cyclic, the circulation in any circuit embracing the origin being 2irfjL. 70 Motion of a Liquid in Two Dimensions [chap, iv 2°. Let us take W = — fJb log z — a z + a (3) If we denote by r 1} r 2 the radii drawn to any point in the plane xy from the points (+ a, 0), and by 9 ly 6 2 the angles which these radii make with the positive direction of the axis of x, we have z — a = ri e^ 1 , z + a = r 2 e i02 , whence <$> — — f^logr^^, ^ = — ^(^1 — ^2) (4) The curves $ = const., ^ = const, form two orthogonal systems of 'coaxal' circles. Either of these systems may be taken as the equipotential curves, and the other system will then form the stream-lines. In either case the velocity at the points (± a, 0) will be infinite. If these points be accordingly isolated by drawing closed curves round them, the rest of the plane xy becomes a triply-connected region. If the circles X — 6 2 = const, be taken as the stream-lines we have the case of a source and a sink, of equal intensities, situate at the points (+ a, 0). If a is diminished indefinitely, whilst /xa remains finite, we reproduce the assumption of Art. 60 (5), which corresponds to the case of a double line-source at the origin. The lines of motion are shewn (in part) on p. 76. If, on the other hand, we take the circles ?\/r 2 = const, as the stream-lines we get a case of cyclic motion, viz. the circulation in any circuit embracing 64] A How of Sources 71 the first (only) of the above points is 27r/x, that in a circuit embracing the second is - 27t/a; whilst that in a circuit embracing both is zero. This example will have additional interest- for us when in Chapter VII. we come to treat of ' Rectilinear Vortices.' 3°. By a simple combination of sources we can represent the flow past a circular barrier due to a source at a given external point P. Let Q be the inverse point of P with respect to the circle, and imagine equal sources p at — X P and Q, and a sink - \i at the centre 0. Then, referring to (2) above, the value of ^ at a point R on the circumference is ^=-lL{RPX+RQX-R0X)=- i L(RPX+0RQ)=- l x(RPX+RP0)=-7rn, a constant over the circle*. 4°. The potential- and stream-functions due to a row of equal and equidistant sources at the points (0, 0), (0, ±a), (0, ±2a), ... are given by the formula w oc log z + log (z - ia) + log (z - ia) + log (z - 2ia) + log (z + 2ia) + . . ., (5) or, say, w = (71ogsinh^- , (6) where C is real. This makes , 1 m 1/ i^ 71 *- 27 2?r #\ / m. i ftan(iry/a) 1 d> = - C log -cosh cos— & , \^=<7tan-Mr — u , , \ \ ^2 & 2\ a a J r (tanh (nx a)) •(7) in agreement with a result given by Maxwell t. The formulae apply also to the case of a source midway between two fixed boundaries y= ±^a. The case of a row of double sources having their axes parallel to x is obtained by- differentiating (6) with respect to z. Omitting a factor we have w = Ccoth — , (8) Csinh(27rx/a) _ Csm(27ry/a) ^ — cosh (27rx/a)- cos (fliry/a) 3 ^ cosh (2nx/a) - cos (2rry/a) ' * ^ ' Superposing a uniform motion parallel to x negative, we have w = z + C coth — , (10) Cainh. (2scxJa) . C sin (2iry Id) ,,,. qji (b=x-\- - - - vs = v ^ oil ni^ r cosh (ZTrxja) - cos {2iry\a) ' y ^ cosh (2irx/a) — cos (Siry/a) ' *" v ; The stream-line ^ = now consists in part of the line y = 0, and in part of an oval curve whose semi-diameters parallel to x and y are given by the equations 8mn 2!^ = I^ y tan- y = a (12) * Kirchhoff, Pogg. Ann., lxiv. (1845) [Ges. Abh. 1]. f Electricity and Magnetism, Art. 203. 72 Motion of a Liquid in Two Dimensions [chap, iv If we put <7=7r6 2 /a, (13) where b is small compared with a* these semi-diameters are each equal to b, approxi- mately. We thus obtain the potential- and stream-functions for a liquid flowing through a grating of parallel cylindrical bars of small circular section. The second of equations (11) becomes in fact, for small values of x, y, M 1 -;^?) (14) 65. If w be a function of z, it follows at once from the definition of Art. 62 that z is a function of w. The latter form of assumption is sometimes more convenient analytically than the former. The relations (1) of Art. 62 are then replaced by dx = dy_ fa dy £ d4> fyr' df a<£ K } A , dw dd> dyjr Also since -y- - = K - + % ^- = — u 4- iv, dz doc dx i dz 1 we have — — = - dw u — iv q\q qJ where q is the resultant velocity at (x, y). Hence if we write <>-£■ (2) and imagine the properties of the function f to be exhibited graphically in the manner already explained, the vector drawn from the origin to any point in the plane of f will agree in direction with, and be in magnitude the reciprocal of, the velocity at the corresponding point of the plane of z. Again, since 11 q is the modulus of dz/dw, i.e. of dx/d<f> + idy/d<j>, we have K-f)'+© ! <*> which may, by (1), be put into the equivalent forms (4) The last formula, viz. -, S |M , (5) expresses the fact that corresponding elementary areas in the planes of z and w are in the ratio of the square of the modulus of dz/dw to unity. * The approximately circular form holds however for a considerable range of values of C. Thus if we put C=±a, we find from (12) x/a =-254, y/a = -250. The two diameters are very nearly equal, although the breadth of the oval is half the interval between the stream-lines y= =*=£a. 64-66] Inverse Methods 73 66. The following examples of this procedure are important. 1°. Assume z — ccoshw, (1) or x — c cosh <f> cos -\jr,) .^\ y = c sinh <f> sin yfr. J The curves <£> = const, are the ellipses ^ i V __ -I (o\ c 2 cosh 2 (/> c 2 sinh 2 </> ' * v ; and the curves yjr = const, are the hyperbolas _^ l^_ = l (4) C 2 cos 2 yjr c 2 sin 2 yfr ' these conies having the common foci (+ c, 0). The two systems of curves are shewn below. Since at the foci we have = 0,-^ = rnr, n being some integer, we see by (2) of the preceding Art. that the velocity there is infinite. If the hyperbolas be taken as the stream-lines, the portions of the axis of at which lie outside the points (± c, 0) may be taken as rigid boundaries. We obtain in this manner the case of a liquid flowing from one side to the other of a thin plane partition, through an aperture of breadth 2c. 74 Motion of a Liquid in Two Dimensions [chap, iv If the ellipses be taken as the stream-lines we get the case of a liquid circulating round an elliptic cylinder, or, as an extreme case, round a lamina whose section is the line joining the foci (+ c, 0). At an infinite distance from the origin <j> is infinite, of the order logr, where r is the radius vector; and the velocity is infinitely small of the order 1/r. 2°. Let z = w + e w , (5) or x = (/> + e* cos -\jr, y = i/r + e^ sin yjr (6) The stream-line ty = coincides with the axis of x. Again, the portion of the line y = tt between x = — oo and x = — 1, considered as a line bent back on itself, forms the stream -line ^r = ir; viz. as <j> decreases from -+- oo through to — oo , x increases from — oo to — 1 and then decreases to — oo again. Similarly for the stream-line -vjr = — tt. Since f = — dzjdw = — 1 — efi cos ty — id> sin \^, it appears that for large negative values of (f> the velocity is in the direction of ^-negative, and equal to unity, whilst for large positive values it is zero. The above formulae therefore express the motion of a liquid flowing into a canal bounded by two thin parallel walls from an open space. At the ends of the walls we have </> = 0, \/r => ± tt, and therefore f=0, i.e. the velocity is 66-67] General Formulae 75 infinite. The forms of the stream-lines, drawn, as in all similar cases in this chapter, for equidistant values of ^, are shewn in the figure on p. 74*. If the walls instead of being parallel make angles ±/3 with the line of symmetry, the appropriate formula is z=l—0: (l-e-»«')+e( 1 - n > w } (7) n where »=£/*-. The stream-lines \^=±tt follow the course of the walls t. This agrees with (5) when n tends to the limit 0, whilst if n = \ we have virtually the case shewn on p. 73. If we change the sign of w in (5) the direction of flow is reversed. If we further super- pose a uniform stream in the negative direction of %, by writing w-z for w, we obtain { w = e z ~ w , or z = w + \ogiv (8) The velocity between the walls at a great distance to the left is now annulled, and we have an idealized representation of a Pitot tube (Art. 24). The stream-lines can be plotted from the formulae .z = (£+*log((£ 2 + xP), .y = ^ + tan" 1 (W) (9) 67. It is known that a function f(z) which is finite, continuous, and single-valued, and has its first derivative finite, at all points of the space included between two concentric circles about the origin, can be expanded in the form f(z) = A + A 1 z+A 2 z*+...+B 1 z- 1 + B 2 z- 2 + (1) If the above conditions be satisfied at all points within a circle having the origin as centre, we retain only the ascending series ; if at all points without such a circle, the descending series, with the addition of the constant A , is sufficient. If the conditions be fulfilled for all points of the plane xy without exception,/ (z) can be no other than a constant A . Putting /(y)=<£+^, introducing polar co-ordinates, and writing the complex constants A nt B n in the forms P n +iQn, R n + i8n, respectively, we obtain- ed = P + 2T r n (P n cos nO - Q n sin n&) + 2? r~ n (R n cos nd + £ n sin nd\] yfr= Q + 2f r n (Q n cos n6 - P n sin nd) + 25° r~ r > (S n cos nO -R n sin n6).\ ' ' These formulae are convenient in treating problems where we have the value of cj), or of d(f>/dn, given over concentric circular boundaries. This value may be expanded for each boundary in a series of sines and cosines of multiples of 6, by Fourier's theorem. The series thus found must be equi- valent to those obtained from (2); whence, equating separately coefficients of sin n6 and cos n6 y we obtain equations to determine P n , Q n , R n , S n . * This example was given by Helmholtz, Berl. Monatsber. April 23, 1868 [Phil. Mag. Nov. 1868 ; Wiss. Abh. i. 154]. t K. A. Harris, "On Two-Dimensional Fluid Motion through Spouts composed of two Plane Walls," Ann. of Math. (2), ii. (1901). A diagram is given for the case of j3 = $Tr. X Kayleigh, Proc. Roy. Soc. A, xci. 503 (1915) [Papers, vi. 329], where a few of the stream- lines are traced. 76 Motion of a Liquid in Two Dimensions [chap, iv 68. As a simple example let us take the case of an infinitely long circular cylinder of radius a moving with velocity U perpendicular to its length, in an infinite mass of liquid which is at rest at infinity. Let the origin be taken in the axis of the c}dinder, and the axes of x, y in a plane perpendicular to its length. Further let the axis of x be in the direction of the velocity U. The motion, supposed originated from rest, will necessarily be irrotational, and </> will be single-valued. Also, since fd(j>/dn.ds, taken round the section of the cylinder, is zero, yjr is also single-valued (Art. 59), so that the formulae (2) apply. Moreover, since dcf)/dn is given at every point of the internal boundary of the fluid, viz. — ■=— = U cos 6. for r — a. dr (3) and since the fluid is at rest at infinity, the problem is determinate, by Art. 41. These conditions give P n = 0, Q n = 0, and Ucos6 = ST nor 71 - 1 (R n cos nO + S n sin nO), which can only be satisfied by making Rx = Ua z , and all the other coefficients zero. The complete solution is therefore Ua 2 a Ua2 a <b — — — cos 6, r t- sin#. ■{*) The stream-lines yjr = const, are circles, as shewn above. Comparing with Art. 60 (6) we see that the effect is that of a double source at the origin. 68] Motion of a Cylinder 77 The kinetic energy of the liquid is given by the formula (2) of Art. 61, viz. 2T = p Udyjr = P U 2 a 2 rcos 2 ed6 = M'U 2 } (5) if M', = ira 2 p, be the mass of fluid displaced by unit length of the cylinder. This result shews that the whole effect of the presence of the fluid may be represented by an addition M ' to the inertia per unit length of the cylinder. Thus, in the case of rectilinear motion, if we have an extraneous force X per unit length acting on the cylinder, the equation of energy gives or (M+M') d ^=X, (6) where M represents the mass of the cylinder itself. Writing this in the form at at we learn that the pressure of the fluid is equivalent to a force —M'dU/dt per unit length in the direction of motion. This vanishes when U is constant. The above result can be verified by direct calculation. By Art. 20 (7), (8) the pressure is given by the formula «.a \-if + F(o, ( 7) p vt provided q denotes the velocity of the fluid relative to the axis of the moving cylinder. The term due to the extraneous forces (if any) acting on the fluid has been omitted ; the effect of these would be given by the rules of Hydrostatics. We have, for r = a, ^t = a ^GOS0 f ? a = 4^ 2 sin 2 0, (8) whence p=p (a — cob 0-2U 2 sin 2 0+F(t)\ (9) The resultant force on unit length of the cylinder is evidently parallel to the initial line 0=0; to find its amount we multiply by —add. cos and integrate with respect to 6 between the limits and 2n. The result is -M'dU/dt, as before. If in the above example we impress on the fluid and the cylinder a velocity — U we have the case of a current flowing with the general velocity U past a fixed circular cylinder. Adding to cj> and ^r the terms Ur cos 6 and Ur sin 0, respectively, we get $= U L.+ -)cos<9, ^= U(r- -) sin 6> (10) The stream-lines are shewn on the next page. If no extraneous forces act, and if U be constant, the resultant force on the cylinder is zero. Cf. Art. 92. 78 Motion of a Liquid in Two Dimensions [chap, iv 69. To render the formula (1) of Art. 67 capable of representing any case of continuous irrotational motion in the space between two concentric circles, we must add to the right-hand side the term ^-log* (1) If A = P + iQ, the corresponding terms in <fi, ty are P\ogr-Qd, P0 + Qlogr, (2) respectively. The meaning of these terms is evident ; thus 2irP, the cyclic constant of -v/r, is the flux across the inner (or outer) circle ; and 2irQ, the cyclic constant of </>, is the circulation in any circuit embracing the origin. For example, returning to the problem of the last Art., let us suppose that in addition to the motion produced by the cylinder we have an independent circulation round it, the cyclic constant being k. The boundary-condition is then satisfied by <f> = U — cos 6 T r 6. \ir (3) The effect of the cyclic motion, superposed on that due to the cylinder, will be to augment the velocity on one side, and to diminish (and, it may be, to reverse) it on the other. Hence when the cylinder moves in a straight line with constant velocity, there will be a diminished pressure on one side, and an increased pressure on the other, so that a constraining force must be applied at right angles to the direction of motion. 69] Cylinder with Circulation 79 The figure shews the lines of flow. At a distance from the origin they approximate to the form of concentric circles, the disturbance due to the cylinder becoming small in com- parison with the cyclic motion. When, as in the case represented, {7>K/2xra, there is a point of zero velocity in the fluid. The stream -line system has the same configuration in all cases, the only effect of a change in the value of U being to alter the radius of the cylinder on the scale of the diagram. When the problem is reduced to one of steady motion we have in place of (3) 4=u(r + ^coa0-£-6, (4) whence - = const. — \ a' 2 P = const. -U 2U sin + cp— ) , (5) for r = a. The resultant pressure on the cylinder is therefore r2tr — / p sin 6 a d6 — + <p U, (6) at right angles to the general direction of the stream. This result is independent of the radius of the cylinder. It will be shewn later that it holds for any form of section*. To calculate the effect of the fluid pressures on the cylinder when moving in any manner we may conveniently adopt moving axes, the origin being taken at the centre^ and the axis of x in the direction of the velocity U. If x De the angle which this makes with a fixed direction, the equation (6) of Art. 20 gives p~ dt ^ q dt dd ' { ' * This important theorem is due to Kutta and Joukowski; see Kutta, Sitzb. d. h. bayr. Akad. d. Wiss. 1910. Proofs are given later (Arts. 72 b, 372). 80 Motion of a Liquid in Two Dimensions [chap, iv where q now denotes fluid velocity relative to the origin, to be calculated from the relative velocity-potential (f>+ Ur cos 6, <p being given by (3). We find, for r=a, f-f-'-i^'-'+^-^'+sS < 8 > The resultant pressures parallel to x and y are therefore /27T J ' TJ /■ 2JT ■% pcos0ad6=-M'-j- t , - psm3ad0 = K P U-M'U^, (9) where M' = irpa 2 as before. Hence, if P, Q denote the components of the extraneous forces, if any, acting on the cylinder in the directions of the tangent and the normal to the path, respectively, the equations of motion of the cylinder are (Jf+JfO-5-P, ) , \ (10) iM+M^uSjt-KpU+Q.) If there be no extraneous forces, U is constant, and writing dx/dt = U/R, where R is the radius of curvature of the path, we find R=(M + M')U/k P (11) The path is therefore a circle, described in the direction of the cyclic motion* If £, t) be the Cartesian co-ordinates of a point on the axis of the cylinder relative to fixed axes, the equations (10) are equivalent to {M+M'y^-Kptj+XA (M+Myr}= <pi+r,j K } where X, Y are the components of the extraneous forces. To find the effect of a constant force, we may put X=(M+M')g' i T=0 (13) The solution then is £ = a + ccos (nt + €) t \ q > (14) 17= /8 +Z-t+c sin (nt+e\ where a, ft, c, e are arbitrary constants, and n = <p/(M+M') (15) This shews that the path is a trochoid, described with a mean velocity g'/n perpendicular to x t. It is remarkable that the cylinder has on the whole no progressive motion in the direction of the extraneous force. In the particular case c = its path is a straight line perpendicular to the force. The problem is an illustration of the theory of ' gyrostatic systems,' to be referred to in Chapter vi. 70. The formula (1) of Art. 67, as amended by the addition of the teim A log z, may readily be generalized so as to apply to any case of irrotational motion in a region with circular boundaries, one of which encloses all the rest. In fact, for each internal boundary we have a series of the form Abg(,-o) + ^L + ( -^ + ..... (1) * Bayleigh, "On the Irregular Flight of a Tennis Ball," Mess, of Math. vii. (1878) [Papers, i. 344]; Greenhill, Mess, of Math. ix. 113 (1880). t Greenhill, I.e. 69-7o] Transformations 81 where c, = a + ib say, refers to the centre, and the coefficients A, A ly A 2 , ... are in general complex quantities. The difficulty however of determining these coefficients so as to satisfy given boundary conditions is now so great as to render this method of very limited application. Indeed the determination of the irrotational motion of a liquid subject to given boundary conditions is a problem whose exact solution can be effected by direct processes in only a limited number of cases. When the boundaries consist of fixed straight walls, a method of transformation devised by Schwarz * and Christoffelf , to be explained in Art. 73, is available. Most of the problems however whose solution is known have been obtained by an inverse method, viz. we take some known form of or yfr and inquire what boundary conditions it can be made to satisfy. Some simple examples of this procedure have already been given in Arts. 63, 64. If we take a known problem of flow with given fixed boundaries, where w=f(z), say, and apply a conformal transformation z = x( z ')> the transformed boundaries in the plane of z will still be stream -lines, and in this way we derive the solution of a new problem. It is sometimes advantageous to effect the transformation in two or more successive steps. A problem which has led to important transformations in this way is that of the flow past a fixed circular cylinder. It is easily seen from Arts. 68, 69 that the general solution of this is w=£ ^ + ^_ iF (*-f) + |logi, (2) where — U, — V are the component velocities at infinity, and k is the circulation. The procedure followed is to write z=t + c, (3) where t is an intermediate complex variable and | c |<a, and finally b 2 *' = ' + 7 (4) It is obvious that the infinitely distant regions of the planes z and z' will be identical, and the general direction of the stream, and the value of the circulation, therefore the same. The constants c and b are adjusted so that the points t=±b in the plane of t may corre- spond to two arbitrary points A, B in the plane of z. For instance, let AB be a chord of the circle r = a, parallel to Ox and subtending an angle 2/3 at the centre 0. Referring to the figure on the next page we find c=-mcos/3, 6 = asin/3 (5) * "Ueber einige Abbildungsaufgaben," Crelle, lxx. [Gesammelte Abhandlungen, Berlin, 1890, ii. 65]. t "Sul problema delle temperature stazionarie e la rappresentazione di una data superficie," Ann. di. Mat. (2) i. 89. See also Kirchhoff, " Zur Theorie des Condensators," Berl. Monatsber. 1877 [Ges. Abh. 101]. Many of the solutions which can thus be obtained have interesting applica- tions in Electrostatics, Heat-Conduction, &c. See, for example, J. J. Thomson, Recent Researches in Electricity and Magnetism, Oxford, 1893. 82 Motion of a Liquid in Two Dimensions [chap, iv Then if P be any other point in the plane of z we have z = OP, t=CP (6) It follows from (4) that i^ = 0^|V (7) Writing for a moment t-b = r l e i0 ^ 1 t + b=r 2 e i \ z'-2b=r 1 'e ie i i z'+2b = r 2 'e i0 *\ (8) we have 6{-6{=Z (0, - 2 ) (9) Z' Now let P describe the circle in the plane of 0, in the positive direction, starting from A. The corresponding point P' in the plane of z' will, by (9), move so that the angle A'P'B' is constant and equal to 2/3, the path therefore being an arc of a circle. As P passes B, B 2 increases by ir ; hence in order that the equation (9) may subsist, 6 2 must increase by 2tt. Hence as P completes its circle, P' moves back again along the arc B'A'. We thus obtain the case of a stream flowing in an arbitrary direction and with arbitrary circulation past a cylindrical lamina whose section is an arc of a circle* S-S/O-?). < 10 > the velocity at the edges A', B' will be infinite. It can be made finite, however, at one edge, say B', by a suitable determination of the circulation, viz. K =47ra(tf r cos/3- F sin/3) (11) The flow at B' is then given by u-iv=(Usinp+ V cos 0) sin pe 2ifi , (12) and is of course tangential to the arc. If the general velocity of the stream is W, at an inclination a to B'A', we have C7=-Fcosa, F=-Tfsina (13) Also, if R is the radius of the arc, asin/3 = i2sin2/3 (14) The ' lift,' therefore, at right angles to the stream, as given by Art. 72 b, is 4 ^ 2 ^Sf cos(a+£) (15) If instead of the circle r = a in the figure we take as the circle to be transformed a circle touching it at A, and just including B, we get the profile of a Joukowsky aerofoil, of * Kutta, I.e. ante p. 79. Some related problems are discussed by Blasius, Zeitschr. f. Math. u. Phys. lix. 225 (1911). 7o-7 i] General Problem of Translation 83 which the circular arc is, as it were, the skeleton* This has a cusp at the point corre- sponding to A, and so involves an infinite velocity at this point (only). This singularity may be avoided by giving a suitable value to k. A simple method of obtaining solutions in two important cases of two- dimensioned motion is explained in the following Arts. 71. Case I. The boundary of the fluid consists of a rigid cylindrical surface which is in motion with velocity U in a direction perpendicular to the length. Let us take as axis of x the direction of this velocity U, and let 8s be an element of the section of the surface by the plane xy. Then at all points of this section the velocity of the fluid in the direction of the normal, which is denoted by dyjr/ds, must be equal to the velocity of the boundary normal to itself, or — Udy/ds. Integrating along the section, we have ijr = — Uy 4- const (1) If we take any admissible form of yjr, this equation defines a system of curves each of which would by its motion parallel to x give rise to the stream-lines yfr = const, f. We give a few examples. 1°. If we choose for yjr the form — Uy, (1) is satisfied identically for all forms of the boundary. Hence the fluid contained within a cylinder of any shape which has a motion of translation only may move as a solid body. If, further, the cylindrical space occupied by the fluid be simply-connected, this is the only kind of irrotational motion possible. This is otherwise evident from Art. 40; for the motion of the fluid and the solid as one mass evidently satisfies all the conditions, and is therefore the only solution which the problem admits of. 2°. Let y\r = Ajr . sin 6; then (1) becomes A r sin# = — Ur sin 6 = const (2) In this system of curves is included a circle of radius a, provided Aja = — Ua. Hence the motion produced in an infinite mass of liquid by a circular cylinder moving through it with velocity U perpendicular to its length, is given by Ua 2 . A ^ = -— sintf, (3) which agrees with Art. 68. * For further developments, and modifications of the method, reference may be made to Glauert, Aerofoil and Airscrew Theory, Cambridge, 1926. t Cf. Eankine, I.e. ante p. 63, where the method is applied to obtain curves resembling the lines of ships. 84 Motion of a Liquid in Two Dimensions [chap, iv 3°. Let us introduce the elliptic co-ordinates f , 77, connected with x, y by the relation x-\-iy — o cosh (f + ir)), (4) or x = c cosh f cos 77,) y = c sinh f sin t?*] ' ^ ' (cf. Art. 66), where f may be supposed to range from to 00 , and 77 from to 27r. If we now put + t> = Ce-G+*fl, (6) where is some real constant, we have yfr = — Ce~Z sin 77, (7) so that (1) becomes Ce - ^ sin rj=Uc sinh f sin 7; + const. In this system of curves is included the ellipse whose parameter £ is determined by Ce~Zo = £/csinh f . If a, b be the semi-axes of the ellipse we have a = c cosh f , 6 = c sinh £ , so that = r = lib r . a — 6 \a — b/ Hence the formula yjr=z-JJb( j) e~^sinr) (8) 71] Translation of an Elliptic Cylinder 85 gives the motion produced in an infinite mass of liquid by an elliptic cylinder of semi-axes a, 6, moving parallel to the greater axis with velocity U. That the above formulae make the velocity zero at infinity appears from the consideration that, when f is large, Bx and By are of the same order as e^SI; and e^Brj, so that dyfr/dx, dyfr/dy are of the order e~ 2 % or 1/r 2 , ultimately, where r denotes the distance of any point from the axis of the cylinder. At infinity yfr tends to the form A sin 6/r as in the case of a double source. If the motion of the cylinder were parallel to the minor axis, the formula would be ir^Va^^fe-ioosv (9) The stream-lines are in each case the same for all confocal elliptic forms of the cylinder, so that the formulae hold even when the section reduces to the straight line joining the foci. In this case (9) becomes yjr= VceScosr), (10) which would give the motion produced by an infinitely long lamina of breadth 2c moving 'broadside on' in an infinite mass of liquid. Since however this solution makes the velocity infinite at the edges, it is subject to the practical limitation already indicated in several instances*. The kinetic energy of the fluid is given by 2T = p Udylr = pC 2 e- 2 £ol 2n cos 2 7) drj = Trpb 2 U\ (11) where b is the half-breadth of the cylinder perpendicular to the direction of motion. Where there is circulation k round the cylinder we have merely to add to the above values of ojr a term /cf/27r. In the case of the lamina the value of k may be adjusted so as to make the velocity finite at one edge, but not at both. If the units of length and time be properly chosen we may write for (4) and (6) x+ iy = cosh (£ + irj), <t> + fy = e~ { ^ +il,) , whence x =* ( l+ ^hp) ' *-+ i 1 -0^p) • These formulae are convenient for tracing the curves cf> — const., ^ = const., which are figured on the preceding page. By superposition of the results (8) and (9) we obtain, for the case of an elliptic cylinder having a velocity of translation whose components are U, V, ^ = ~(^r|) e~^ (Ub sin rj-Va cost}) (12) To find the motion relative to the cylinder we must add to this the expression Uy— Vx=c(U sinh £sin?7— Fcosh £cos?7) (13) * This investigation was given in the Quart. Journ. of Math. xiv. (1875). Eesults equivalent to (8), (9) had however been obtained, in a different manner, by Beltrami, "Sui principii fonda- mentali dell' idrodinamica razionale," Mem. deW Accad. delle Scienze di Bologna, 1873, p. 394. [Opere matematiche, Milano, 1904, ii. 202.] 86 Motion of a Liquid in Two Dimensions [chap, iv For example, the stream-function for a current impinging at an angle of 45° on a plane lamina whose edges are at x= ±c is ^=- -^^ocsinh^cosiy-sin^), (14) where q is the velocity at infinity. This immediately verifies, for it makes ^ = for £ = 0, and gives for £=oo . The stream-lines for this case (turned through 45° for convenience) are shewn below. They will serve to illustrate some results to be obtained later in Chapter vi. If we trace the course of the stream-line \fs=0 from <f>= +oo to 0= — oo , we find that it consists in the first place of the hyperbolic arc »/ = J 7r, meeting the lamina at right angles; it then divides into two portions, following the faces of the lamina, which finally re-unite and are continued as the hyperbolic arc ?7 = f 7r. The points where the hyperbolic arcs abut on the lamina are points of zero velocity, and therefore of maximum pressure*. It is plain that the fluid pressures on the lamina are equivalent to a couple tending to set it broadside on to the stream ; and it is easily found that the moment of this couple, per unit length, is ^7rpqd 2 c 2 f. Compare Art. 124. 72. Case II. The boundary of the fluid consists of a rigid cylindrical surface rotating with angular velocity co about an axis parallel to its length. Taking the origin in the axis of rotation, and the axes of x, y in a perpen- dicular plane, then, with the same notation as before, d\fr/ds will be equal to the normal component of the velocity of the boundary, or dyfr _ dr ds ds' * Prof. Hele Shaw has made a number of beautiful experimental verifications of the forms of the stream-lines in cases of steady irrotational motion in two dimensions, including those figured on p. 78 and on this page; see Trans. Inst. Nav. Arch. xl. (1898). The theory of his method will find a place in Chapter xi. t When the general direction of the stream makes an angle a with the lamina the couple is %irpq 2 c 2 sin 2a. Cisotti, Ann. di. mat. (3), xix. 83 (1912). 71-72] Rotating Boundary 87 if r denote the radius vector from the origin. Integrating we have, at all points of the boundary, yfr = £&>r 2 + const (1) If we assume any possible form of yjr, this will give us the equation of a series of curves, each of which would, by rotating round the origin, produce the system of stream-lines determined by \fr. As examples we may take the following: 1°. If we assume yfr = Ar 2 cos 20 = A {x 2 - y 2 ), (2) the equation (1) becomes (iG>-A)x 2 + (ia> + A)y 2 = C, which, for any given value of A, represents a system of similar conies. That this system may include the ellipse a 2 *b 2 ~ ' we must have (\a> — A) a 2 = (\w + A)b 2 , or A = |q> . = . * or + 6 2 a 2 - b 2 Hence the formula \fr = \ w . 2 2 (^g 2 — y 2 ) (3) gives the motion of a liquid contained within a hollow cylinder whose section is an ellipse with semi-axes a, 6, produced by the rotation of the cylinder about its longitudinal axis with angular velocity &>. The arrangement of the stream-lines yfr = const, is shewn on the next page. The corresponding formula for <j> is a 2 — b 2 +—»-#+v-*y (*) The kinetic energy of the fluid, per unit length of the cylinder, is given by This is less than if the fluid were to rotate with the boundary, as one rigid mass, in the ratio of f a 2 - b 2 \ 2 U 2 + 6V to unity. We have here an illustration of Lord Kelvin's minimum theorem, proved in Art. 45. 2°. With the same notation of elliptic co-ordinates as in Art. 71, 3°, let us assume <j> + i,lr = Cie- 2 ti +il » (6) Since oc 2 -\-y 2 = \c 2 (cosh 2f + cos 2rj), the equation (1) becomes Ce~^ cos 2rj — \wc 2 (cosh 2f + cos 2r)) = const. 88 Motion of a Liquid in Two Dimensions [chap, iv This system of curves includes the ellipse whose parameter is f , provided or, using the values of a, b already given, a=Jo)(a + 6) 2 , so that yfr = \(o (a + b) 2 e~ 2 ^ cos 2rj, cf> = lo»(a + b) 2 e- 2 tsm2 v .\ ^ At a great distance from the origin the velocity is of the order 1/r 3 . The above formulae therefore give the motion of an infinite mass of liquid, otherwise at rest, produced by the rotation of an elliptic cylinder about its axis with angular velocity co* The diagram shews the stream-lines both inside and outside a rigid elliptical cylindrical case rotating about its axis. The kinetic energy of the external fluid is given by 2T=l7rpc 4 .a> 2 (8) It is remarkable that this is the same for all confocal elliptic forms of the section of the cylinder. Combining these results with those of Arts. 66, 71 we find that if an elliptic cylinder be moving with velocities U, V parallel to the principal axes of its cross-section, and rotating with angular velocity <w, and if (further) the * Quart. Journ. Math. xiv. (1875) ; see also Beltrami, I.e. ante p. 85. 72] Rotating Cylinder 89 fluid be circulating irrotationally round it, the cyclic constant being te, then the stream-function relative to the aforesaid axes is yfr = - /(5Lt|W* ( Ub sin v - Va cos V ) + J« (a + 6) 2 <r 2 * cos 2t; + ^- f . (9) The ^>a^5 followed by the particles of fluid in several of the preceding cases, as distinguished from the stream-lines, have been studied by Prof. W. B. Morton*; they are very remarkable. The particular case of the circular cylinder (Art. 68) was examined by Maxwell")*. 3°. Let us assume \js = Ar 3 cos 30 = A(x 3 - Zxy 2 ). The equation (1) of the boundary then becomes A (x 3 -3xy 2 )-%a>(x 2 +f) = C. (10) We may choose the constants so that the straight line x=a shall form part of the boundary. The conditions for this are ^a 3 -£o>a 2 = <7, 34a+£a> = 0. Substituting in (10) the values of A, C hence derived, we have x 3 - a 3 - Zxy 2 + 3a (x 2 - a 2 +y 2 ) = 0. Dividing out by x - a, we get x 2 + 4ax + 4a 2 - 3y 2 , or x + 2a=± f J3.y. The rest of the boundary consists therefore of two straight lines passing through the point ( - 2a, 0), and inclined at angles of 30° to the axis of x. We have thus obtained the formulae for the motion of the fluid contained within a vessel in the form of an equilateral prism, when the latter is rotating with angular velocity to about an axis parallel to its length and passing through the centre of its section; viz. we have ^-^-r 3 cos 30, (^=^-^8^30, (11) where 2 v/3a is the length of a side of the prism J. 4°. In the case of a liquid contained in a rotating cylinder whose section is a circular sector of radius a and angle 2a, the axis of rotation passing through the centre, we may assume COS 20 /r\ (2n+l)7r/2a -r/J ^=i^ 2 ^ + ^ 2n + 1 (M cos(2* + l)£, (12) cos 2a ' n + 1 \aJ v T ; 2a the middle radius being taken as initial line. For this makes \js — \ ar 2 for 6 — ± a, and the constants A 2n+1 can be determined by Fourier's method so as to make yjr—^coa 2 for r=a. We find ^n + i-(-)- + 1 a>a 2 { (2 ^^_ 4a - ( ^^ + ( ^ + l ^ + 4a } (13) The conjugate expression for <$> is , , 2 sin20 . / r \ (2/i+D w /2a . n Q ♦--^SHT^-^-W ^(2^+1)- (14) * Proc. Boy. Soc. A, lxxxix. 106 (1913). f Proc. Lond. Math. Soc. iii. 82 (1870) [Papers, ii. 208]. X The problem of fluid motion in a rotating cylindrical case is to a certain extent mathe- matically identical with that of the torsion of a uniform rod or bar. The examples numbered ' 1° ' and '3°' are mere adaptations of two of de Saint- Venant's solutions of the latter problem. See Thomson and Tait, Art. 704 et seq. 90 Motion of a Liquid in Two Dimensions [chap, iv The kinetic energy is given by 2T= - p U^ds= -2p<o ("fardr, (15) where cp a denotes the value of cp for B — a, the value of dcp/dn being zero over the circular part of ihe boundary*. The case of the semicircle <x = \tt will be of use to us later. We then have ^■W M 7fe-W + Ri' (16) and therefore f a 6 rdr=°^2 -i-|-i 2 , l } = «* U «*\ J Va rr 2n + 3\2n-l 2n + l^ 2n + ZJ tt \ 8/' Hencet 2T=$irp<o 2 a i C^-^ = '3106a 2 x|7rpa> 2 a 2 (17) This is less than if the fluid were solidified, in the ratio of 6212 to 1. Cf. Art. 45. 72 a. We have seen in several instances that when a cylinder has a motion of translation though an infinite fluid the effect at a great distance is that of a double source. A general formula for this can be given in terms of certain constants which occur in the expression for the kinetic energy of the fluid j. If we write <p=Ufa + Vfa, (1) where ( U, V) is the velocity of the cylinder, the functions fa , fa are determined by the conditions that V 2 fa = 0, v 2 2 = O throughout the external space, that their derivatives vanish at infinity, and that at the contour of the cylinder _dp -*tt-n, (2) on on where (I, m) is the direction of the outward normal. Hence the energy of the fluid is given by — = - L^ ds=AU 2 + 2HUV+BV 2 , (3) p J on where A = - I fa -^ ds= llfads, \ B= — I fa ~-ds= jmfads, V H= — J fa -+^ds= - I fa ~ds= lmfads= Jlfads. The two forms of E are equal by the two-dimensional form of Green's Theorem. Cf. Art. 121, where the general three-dimensional case is discussed. Referring to Art. 60 (7), suppose that a cylinder of any form of section is moving with unit velocity parallel to the axis of x. Taking an origin within the contour, and writing r 2 = {xQ~xf + {y -yf = r 2 - 2(^o+3/3/o) +•••> ( 5 ) * This problem was first solved by Stokes, "On the Critical Values of the Sums of Periodic Series," Camh. Trans, viii. (1847) [Papers, i. 305]. See also Hicks, Mess, of Math. viii. 42 (1878) ; Greenhill, ibid. viii. 89, and x. 83. t Greenhill, I.e. J Cf. Proc. Boy. Soc. A, cxi. 14 (1926) and Art. 300 infra. •(4) 2-72 b] Source due to a Moving Cylinder 91 where (x 0i y ) is a distant point at which the value of cj> is required, and (#, y) a point of the contour, we have log,=logr -^ + »+... > (6) and _ (log ,. )= _^_ r A0 ) approximately. Writing <£ = $!, <p'=-x (7) in the formula referred to, we find (A + Q)x Q +Hyp /fi v ^ p== V ' ^ where 4 and 5" are defined by (4), and Q=jlxds, (9) i.e. Q denotes the sectional area of the cylinder. The flow at a great distance is accordingly that due to a double source, but the axis of the source does not in general coincide with the direction of motion of the cylinder. The generalization of (8) is obvious. When the cylinder has a velocity ( £7, V) we have 2*r *<t> P ={(A + Q) U+ffV}x + {HU+(B+Q)V}y () (10) In terms of the complex variables w, z, this may be written W = (a + i(3)/z , (11) with 2rra = (A + Q)U+HV, 2irP=HU+(B+Q)V. (12) For an elliptic cross-section we have, by comparison of Art. 71 (11) with (3) above, A = 7rb 2 , B = 7ra 2 , whilst Q = nab. Hence <!> P =(a + b)(bUxo + aVy Q )l2r * (13) 72 b. The hydrodynamic forces on a fixed cylinder due to the steady irrotational motion of a surrounding fluid have already been calculated in one or two cases. A general method, available whenever the form of w, — (f> + iyfr, for the fluid motion is known, has been given by Blasius *. The pressures on the contour may be reduced to a force (X, Y) at the origin, and a couple N. If 6 be the angle which the velocity q makes with the axis of x, we have Y + iX = - ipjq 2 (cos - ism6)ds, (1) where the integral is taken round the contour of the cylinder. This may be written Y+iX^-yfiqe-vf^ds^-ipf^jdz, (2) This gives X and Y. Again, if S- be the angle which an element Bs of the contour makes with the radius vector (produced), N = fpr cos ^rds=jprdr = - \pj{u 2 + v 2 )(xdx + ydy), (3) * " Funktiontheoretische Methoden in der Hydrodynamik," Zeitschr. f. Math. u. Phys. Iviii. (1910). 92 Motion of a Liquid in Two Dimensions [chap, iv Now along a stream-line we ha,ve.vdx = udy, whence (u - ivf (dx + idy) = (u 2 + v 2 ) (dx - idy) and (u — ivf (x + iy) (dx + idy) = (u 2 + v 2 ) [xdx + ydy + i (y dx - xdy)\, Hence N is given by the real part of the integral ■/( £)'•«•- <« In the case of a cylinder immersed in a uniform stream, with circulation, the value of w at a great distance tends to the form w = A + Bz + Clog z (5) Since in (4) there are no singularities of the integrand in the space occupied by the fluid, the integral may be replaced by that round an infinite enclosing contour. On this understanding If the stream at infinity is ( U, V), and if k denote the circulation, we have B = -(U-iV), G=-%k\1tt (7) Hence X = icpV, Y=-k P U (8) which is the generalization of the result obtained in Art. 69 for the particular case of a circular section. For the calculation of the moment N the expression in (5) must be carried a stage further. Writing w- A + Bz+C\ogz + — , (9) we have 'g) ! =* + ^ + ^5 + (10) Omitting all the terms which disappear in the case of an infinite contour we have c, /(^) 2 zdz = 2>rri(C 2 -2BD) (11) Substituting the values of B and G from (7), writing D = a + iff, and taking the real part, we find N=2irp{ffU-aV) (12) If by the superposition of a general velocity (— U, — V) the fluid were reduced to rest at infinity, the term D/z in (9) would be due to a translation of the cylinder with this velocity. Hence the values of a, {3 are as given in Art. 72, except that the signs are reversed. Hence (12) gives N= P {(A-B)UV-H(U 2 -V 2 )} (13) Thus for an elliptic section referred to its principal axes N = -irp{a 2 ~b 2 ) UV. (14) 72 b] Blasius' Theorems 93 As a further application of Blasius' formula we may calculate the force on a fixed cylinder due to an external source. We write w= — plog(z — c) +/(«), (15) where the first term represents the source at z = c, say, and f(z) its image in the cylinder, i.e. f{z) is the addition necessary to annul the normal velocity at the contour, due to the source. Hence S--A*™ ™ The contour integral in (2) is now equal to the integral round an infinite contour minus the integral (in the positive sense) round the singularity at z = c. The infinite contour gives a zero result. In the neighbourhood of the singularity the only part of (dwjdz) 2 which need be taken into account is that containing the first power of z — c in the denominator, viz. Ml ^ z-c ' ultimately. Hence Y+iX= - Mppf (c) (17) The form of / (z) for the case of a circular cylinder is already known from Art. 64, 3°. The source being supposed on the axis of x, so that c is real, we have f(z)=-p\og(z-a 2 /c) + p\ogz, (18) f'^-c^y • < 19 > *-^5F^ r =° ( 2 °)* 27r/z 2 a 2 c{c 2 -a 2 ) In the general case, an approximation to the asymptotic form which f(z) assumes, when the distance of the source is great compared with the dimensions of the cross-section, is obtained if we suppose it to represent the effect of a translation of the cylinder with a velocity equal and opposite to that which the source would produce in the neighbourhood, if the cylinder were absent. Thus, the source being still assumed to be on the axis of x, we have from Art. 72 a m jA±3^m, (21) where tf=/»/& Hence f( c ) = JA±^±^lt > (22 ) and therefore x JA + Q)y?P ^ Y—?& (23) Iff(=p 2 /(?) is the acceleration at the position of the origin, in the undisturbed stream, these results may be written X=p(A + Q)f, Y=- P Hf. (24) For a circular section A = 7ra 2 , H—Q, Q=7ra 2 , and the formula (20) is verified, if we neglect terms of the order a 2 \c 2 . A number of elegant applications of Blasius' method, relating to the mutual action of circular cylinders, with circulation, have been made by Cisottit. One of his results may be quoted. A cylinder of radius b is fixed excentrically within a cylindrical tunnel of radius a, and the intervening space is occupied by fluid having a circulation k. The resultant force on the cylinder is towards the nearest part of the tunnel wall, and has the value K 2 d 2 +27ry/{(a + b + d)(a + b-d)(a-b + d)(a-b-d)}, where d is the distance between the axes. * The result is due to Prof. G. I. Taylor. f Rend. d. r. Accad. d. Lincei (6) i. (1925-6). 94 Motion of <% Liquid in Two Dimensions [chap, iv Free Stream- Lines. 73. The first solution of a problem of two-dimensional motion in which the fluid is bounded partly by fixed plane walls and partly by surfaces of constant pressure, was given by Helmholtz*. Kirchhofff and others have since elaborated a general method of dealing with such questions- If the surfaces of constant pressure be regarded as free, we have a theory of jets, which furnishes some interesting results in illustration of Art. 24. Again since the space beyond these surfaces may be filled with liquid at rest, with- out altering the conditions of the problem, we obtain also a number of cases of 'discontinuous motion,' which are mathematically possible with perfect fluids, but whose practical significance is more open to question. We shall return to this point at a later stage (Chap. XI.); in the meantime we shall speak of the surfaces of constant pressure as 'free.' Extraneous forces, such as gravity, being neglected, the velocity must be constant along any such surface, by Art. 21 (2). The method in question is based on the properties of the function ? introduced in Art. 65. The moving fluid is supposed bounded by stream- lines yjr = const., which consist partly of straight walls, and partly of lines along which the resultant velocity (q) is constant. For convenience, we may in the first instance suppose the units of length and time to be so adjusted that this constant velocity is equal to unity. Then in the plane of the function f the lines for which q = 1 are represented by arcs of a circle of unit radius, having the origin as centre, and the straight walls (since the direction of the flow along each is constant) by radial lines drawn outwards from the circumference. The points where these lines meet the circle correspond to the points where the bounding stream-lines change their character. Consider, next, the function log f. In the plane of this function the circular arcs for which q = 1 become transformed into portions of the imaginary axis, and the radial lines into lines parallel to the real axis, since if ^—q~ x e iB we have log£=logi + t0 (1) It remains, then, to determine a relation of the formj log?=/(w), (2) where tv = (f> + iyfr, as usual, such that the rectilinear boundaries in the plane of log f shall correspond to straight lines -v/r = const, in the plane of w. There are further conditions of correspondence between special points, one on the boundary, and one in the interior, of each region, which render the problem determinate. * Loc. cit. ante p. 75. t "Zur Theorie freier Fliissigkeitsstrahlen," Crelle, lxx. (1869) [Ges. Abh. p. 416]. See also his Mechanik, cc. xxi., xxii. % The use of log £, in place of f, is due to Planck, Wied, Ann. xxi. (1884). 73] Free Stream-Lines 95 When the correspondence between the planes of f and w has been established, the connection between z and w is to be found, by integration, from the relation £— {■ (3) aw The arbitrary constant which appears in the result is due to the arbitrary position of the origin in the plane of z. The problem is thus reduced to one of conformal representation between two areas bounded by straight lines*. This is resolved by the method of Schvvarz and Christoffel, already referred tof, in which each area is repre- sented in turn on a half-plane. Let Z (= X + iY) and t be two complex variables connected by the relation ^ = A(a-t)-^'(b-t)-^'(c-t)-yl ir ..., (4) where a, b, c, . . . are real quantities in ascending order of magnitude, whilst a, /3, 7, ... are angles (not necessarily all positive) such that a + /3 + 7 + ... = 2tt; (5) and consider the line made up of portions of the real axis of t with small semi-circular indentations (on the upper side) about the points a, b, c, ... . If a point describe this line from t = — oo to t = -\- oo , the modulus only of the expression in (4) will vary so long as a straight portion is being described, whilst the effect of the clockwise description of the semi-circular portions is to introduce factors e ia , e ifi , e iy , ... in succession. Hence, regarding dZ/dt as an operator which converts 8t into BZ, we see that the upper half of the plane of t is conformably represented on the area of a closed polygon whose exterior angles are a, /3, 7, ...,by the formula Z = A${a-t)-*l" (b -t)-V" (c-t)-*l" ...dt + B, (6) provided the path of integration in the £-plane lies wholly within the region above delimited. When a, b, c, ...,«, (3> 7, ... are given, the polygon is com- pletely determinate as to shape; the complex constants A, B only affect its scale and orientation, and its position, respectively. As already indicated, we are specially concerned with the conformal representation of rectangular areas. If a = @ = y = 8 = %ir, the formula (6) becomes Z = A ) s/{(a-t){b-t){c-t){d-t)\+ B (7) It is easily seen that the rectangle is finite in all its dimensions unless two at least of the points a, 6, c, d are at infinity. The excepted case is the one * See Forsyth, Theory of Functions, c. xx. t See the footnotes on p. 81 ante. 96 Motion of a Liquid in Two Dimensions [chap, iv Z-A\-,? : +U specially important to us; the two finite points may then conveniently be taken to be t = + 1, so that dt_ = A cosh-^ + 5 (8) In particular, the assumption t — cosh -j-, (9) where k is real, transforms the space bounded by the positive halves of the lines F=0, Y=7rk, and the intervening portion of the axis of F, into the upper half of the plane t Cf. Art. 66, 1°. Again, if the two finite points coincide, say at the origin of t, we have Z=AJj + B = A \ogt + B (10) This transforms the upper half of the £-plane into a strip bounded by two parallel straight lines. For example, if t = e z l\ (11) where k is real, these may be the lines F= 0, Y—irk. 74. As a first application of the method in question, we may take the case of a fluid escaping from a large vessel by a straight canal projecting inwards*. This is the two-dimensional form of Borda's mouthpiece, referred to in Art. 24. The boundaries of corresponding areas in the planes of ?, log f, and w, respectively, are easily traced, and are shewn in the figures f. It remains to connect the areas in the planes of log f and w each with the upper half-plane of an intermediate variable t. It appears from equations (8) and (10) of the preceding Art. that this is accomplished by the substitutions log? =4 cosh- 1 * + 5, w=C\ogt + D (1) We have here made the corners A, A' in the plane of log f correspond to t=± 1, and we have also assumed that £ = corresponds to w = — oo , as is evident on inspection of the figures. To specify more precisely the values of the cyclic functions cosh -1 1 and log t we will assume that they both vanish at t = 1, aud that their values at other points in the positive half-plane are determined by considerations of continuity. It follows that when t = — 1 the value of each function will be iir. At the points A' , A in the plane of log f, * This problem was first solved by Helmholtz, I.e. ante p. 75. t The heavy lines correspond to rigid boundaries, and the fine continuous lines to free surfaces. Corresponding points in the various figures are indicated by the same letters. 73-74] B or da's Mouthpiece 97 we have, on the simplest convention, log f =0 and liir, respectively; whence, towards determining the constants in (1), we have = B, 2iir = iwA + B, so that log£=2cosh- 1 £ (2) Again, in the plane of w we take the line //' as the line -\Jr = 0; and if the final breadth of the issuing jet be 26, the bounding stream-lines will be ylr— ±b. We may further suppose that <£ = is the equipotential curve passing through A and A'. Hence, from (1), ib = i7rC + D, -ib = D, 26 1 , ■* w — — log t — ib. IT so that w — — log t — ib (3) It is easy to eliminate t between (2) and (3), and thence to find the relation ? 1 r i 3 A .) K ^ r logt i A' B _ ji J i' B' w A -I' between z and w by integration, but the formulae are perhaps more convenient in their present shape. The course of either free stream-line, say A' I, from its origin at A', is now easily traced. For points of this line t is real and ranges from 1 to 0; we have, moreover, from (2), id = 2 cosh -1 1, or t = cos \Q. Hence, also, from (3), (f> = — log cos \6. (4) Since, along this line, we have d<j>/ds = — q = — 1, we may put </> = — s, where the arc s is measured from A'. The intrinsic equation of the curve is therefore 26 1 in s = — log sec \v. 7T (5) 98 Motion of a Liquid in Two Dimensions [chap, iv From this we deduce in the ordinary way a? = — (sin 2 |0-logsec£0), v = -(0-sin0), (6) 7T IT if the origin be at A'. By giving a series of values ranging from to it, the curve is easily plotted*. Line of Symmetry. Since the asymptotic value of y is b, it appears that the distance between the fixed walls is 46. The coefficient of contraction is therefore J, in accord- ance with Borda's theory. 75. The solution for the case of fluid issuing from a large vessel by an aperture in a plane wall is analytically very similar. The chief difference is that the values of log f at the points A, A' in the figures must now be taken . ?■ _ Ir- J log}; A' A> B' to be and — iir, respectively, whence, to determine the constants A, B in. (1) we have Q^iwA+B, -iir = B, so that log f= cosh -1 t — iir (7) * To correspond exactly with p. 97 the figure should be turned through 180°. 74- '6] Vena Contracta 99 The relation between w and t is exactly as before, viz. w = — log t — ib, (8) 7T where lb is the final breadth of the stream, between the free boundaries. For the stream-line AI, t is real, and ranges from — 1 to 0. Since, also, id = cosh -1 1 — iir we may put t = cos (0 + it), where 6 varies from to — -J7T. Hence, from (8), with </> = — s, we have, for the intrinsic equation of the stream- line, = f£lo g (_ S ec0) (9) 7T From this we find x = - sin 2 ^, y=-{logtan(i7r + l<9)-sin<9}, (10) 7T 7T if the point A in the plane of z be taken as origin*. The curve is shewn (in an altered position) below. Line of Symmetry. The asymptotic value of x, corresponding to 6 = — J7r, is 26/-7T, the half width of the aperture is therefore (it + 2)6/7r, and the coefficient of contraction is 7r/(7r + 2) = '611. 76. In the next example a stream of infinite breadth is supposed to impinge directly on a fixed plane lamina, and thence to divide into two portions bounded internally by free surfaces. The middle stream-line, after meeting the lamina at right angles, branches off into two parts, which follow the lamina to the edges, and thence form the * This example was given by Kirchhoff (I.e.), and discussed more fully by Rayleigh, "Notes on Hydrodynamics," Phil. Mag. Dec. 1876 [Papers, i. 297]. 100 Motion of a Liquid in Two Dimensions [chap, iv free boundaries. Let this be the line yfr = 0, and let us further suppose that at the point of divergence we have <£ = 0. The forms of the boundaries in the various planes are shewn in the figures. The region occupied by the moving logZ w -—i w >r~ fluid now corresponds to the whole of the plane w, which must however be regarded as bounded internally by the two sides of the line yfr — 0, <f> < 0. With the same conventions as in the beginning of Art. 75, we have log f =cosh -1 £ -iir, (1) or *=-cosh(iogo=-i(r-J). ■(2) The correspondence between the planes of w and t is best established by considering first the boundary in the plane of w~\ The method of Schwarz and Christoffel is then at once applicable. Putting a = — 7r, /8 = y = . . . = 0, in Art. 73 (4), we have iv-^iAP + B (3) dt = AL At I we have t = 0, w~ x = 0, so that B = 0, or (say) w = — .(4) To connect C (which is easily seen to be real) with the breadth (I) of the lamina, we notice that along C A we have ? = <f~\ and therefore, from (2), *--*(£ + «). ? = -*-V(« 2 -l), .(5) 76] Impact of a Stream on a Lamina 101 the sign of the radical being determined so as to make q = for t = — oo . Also, dx/d(j> = - 1/q. Hence, integrating along G A in the first figure we have '= 2 /::s^=- 4 Cl=- 4 °/:: w-< ■■■«» whence G= — — r (7) Along the free boundary AI, we have log f= id, and therefore, from (2) and £ = -cos0, </>=-asec 2 (9 (8) The intrinsic equation of the curve is therefore s=-L-,se<*0, (9) 7T + 4 where 6 ranges from to — \tt. This leads to 21 tf= 7jr ^(sec0 + i7r), y = j {sec tan - log (£ir + £0)}, the origin being at the centre of the lamina. .(10) Line of Symmetry. The excess of pressure on the anterior face of the lamina is, by Art. 23 (7), equal to ^p(l — q 2 ). Hence the resultant force on the lamina is p/>-rtg*--^/:| i g-«)j--^/;>-i)$-^ft (11) It is evident from Art. 23 (7), and from the obvious geometrical similarity of the motion in all cases, that the resultant pressure (P , say) will vary as 102 Motion of a Liquid in Two Dimensions [chap, iv the square of the general velocity of the stream. We thus find, for an arbitrary velocity q *, Po=^^pqo 2 .l = -^Op q( ?.l (12) 77. If the stream be oblique to the lamina, making an angle a, say, with its plane, the problem is modified in the manner shewn in the figures. // / '/ < ? , I 4 i' C w A' IV I A' The equations (1) and (2) of the preceding Art. still apply; but at the point I we now have £=e -*(*-«), and therefore £ = cosa. Hence, in place of (4)t, C (13) (if— cos a) 2 ' " At points on the front face of the lamina, we have, since ^ -1 = |^|, 1 = ±*+V(* 2 -l), q=±t- */(?-!), ,(14) where the upper or the lower signs are to be taken according as t% 0, i.e. according as the point referred to lies to the left or right of C in the first figure. Hence dx ,ldcj) 2C ^ =± ^ = (^o^±V(* 2 -l)} (15) Between A' and C, t varies from 1 to oo , whilst between A and C the range is from — oo to -1. If we put 1 - COS O COS 0) t= COS O - COS 0) the corresponding ranges of o> will be from ir to a, and from a to 0, respectively ; and we find dt cos a — cos to . , , //j9 _. sin a sin o) sincoao), ±v(£ 1) = ' (t - cos a) sin* a cos a — cos co * Kirchhoff, I.e. ante p. 94; Kayleigh, "On the Kesistance of Fluids," Phil. Mag. Dec. 1876 [Papers, i. 287]. f The solution up to this point was given by Kirchhoff (Crelle, I.e.) ; the subsequent discussion is taken, with merely analytical modifications, from the paper by Kayleigh. 76-77] Pressure on a Lamina Hence and therefore dx da> 2(7 sin^o — (1 — cos a cos co + sin a sin a>) sin co, 103 .(16) x = . . {2 cos co + cos a sin 2 o + sin a sin co cos co + (h n — co) sin a}, (17) sin 4 a where the origin has been adjusted so that x shall have equal and opposite values when o> = and co = 7r, respectively; i.e. it has been taken at the centre of the lamina. Hence, in terms of C, the whole breadth is A. -4- tt sitl n (18) , _ 4 + 7r sin o ~ sin 4 a The distance, from the centre, of the point (<o=o) at which the stream divides is _2coso(l+sin 2 a) + (|7r-a)sina , »a» 4 -I- it sin a To find the total pressure on the front face, we have \ ? {\-?)dx=±\ P [~ q yj t dt=±^c>](?-\) = — ?™ • snv 2 co da sin 3 a dt (t-cosa) .(20) Integrated between the limits it and 0, this gives 7rpC/sin 3 a. Hence, in terms of I, and of an arbitrary velocity q of the stream, we find Pn = 7r sin a .pqf.l (21) 4 + 7rsina r *° To find the centre of pressure, we take moments about the centre of the lamina. Thus hpl(I—q 2 )xdx= — r-^- . / #sin 2 coc£co 2 W «m 3 a J n rrpC „ C cos a X ^ ; sm°o sin 4 a (22) on substituting the value of x from (17). The first factor represents the total pressure; the abscissa x of the centre of pressure is therefore given by the second, or, in terms of the breadth, _ „ cos a 7 /OQ x a? = f — : — .1 (23) 4 4 + 7rsina In the following table, derived from Rayleigh's paper, the column I gives the excess of pressure on the anterior face, in terms of its value when a = 90° ; whilst columns II and III give respectively the distances of the centre of pressure, and of the point where the stream divides, from the centre of the lamina, expressed as fractions of the total breadth*. a I II III 90° 1-000 •000 •000 70° •965 •037 •232 50° •854 •075 •402 30° •641 •117 •483 20° •481 •139 •496 10° •273 •163 •500 * For a comparison with experimental results see Rayleigh, I.e. and Nature, xlv. (1891) [Papers, iii. 491]. 104 Motion of a Liquid in Two Dimensions [chap, iv 78. An interesting variation of the problem of Art. 76 has been discussed by Bobyleff *. A stream is supposed to impinge symmetrically on a bent lamina whose section consists of two equal straight lines forming an angle. If 2a be the angle, measured on the down-stream side, the boundaries in the plane of £ can be transformed, so as to have the same shape as in the Art. cited, by the assumption provided A and n be determined so as to make £' = 1 when ^=e~ l ^ 1t ~ a \ and (' = e~ llt when C=e" i( ^ +a) . This gives A=e~ i{hn - a) , n = 2a/7r. On the right-hand half of the lamina, t will be negative as before, and since 2"" 1= |t|, i = {-* + V(* 2 -l)}», q = {-t-^-\)Y (24) Hence /^^'"^/"j-^ /j**"^/! 1 "'"*" 1 ^?""^^''"^" 1 * dt *V(* 2 -i)' dt These can be reduced to known forms by the substitution where a ranges from to 1. We thus find 7i\ - ~dt = -l-2?i — — - da>=-I-n-n 2 I — — da>, CJ -«,q dt J (l+(o) 2 Jo l+o) ^ q -£dt = -l + 2n — — rs d<» = - 1 - w + w 2 / — — eta. CJ-ao^dt J o(l+») 2 Jo 1+0) (7 We have here used the formulae .(25) J (l+w) J ol+o) jo(l+o)) 2 2 Jol+o) / where 1 > h > 0. Since, along the stream-line, ds/d(f) = — l/q, we have from (25), if b denote the half- breadth of the lamina, 6= 4 + LV4f£^4 (27) The definite integral which occurs in this expression can be calculated from the formula Irfi^-^pRD+i*^-**)-***-**). ( 28 > where ¥ (m), = d/dm . log n (m), is the function introduced and tabulated by Gauss t. The normal pressure on either half is, by the method of Art. 76, ^jJ-'fLjW^^f^^^rfc j,. ip c.-^-....(29) 2H J -ooXq *J dt 2 r j 1+w 2 r sin^TT r tt sin a v y * Journal of the Russian Physico- Chemical Society, xiii. (1881) [Wiedemann's Beiblatter, vi. 163]. The problem appears, however, to have been previously discussed in a similar manner by M. K6thy, Klausenburger Berichte, 1879. It is generalized by Bryan and Jones, Proc. Roy. Soc. A, xci. 354 (1915). + " Disquisitiones generales circa seriem infinitam... ," Werke, Gottingen, 1870... , iii. 161. 78-79] Bobyleff's Problem 105 The resultant pressure in the direction of the stream is therefore 4a 2 P C. .(30) Hence, for any arbitrary velocity q of the stream, the resultant pressure is P=py.p q0 *b, .(31) where L stands for the numerical factor in (27). For = ^77, we have L = 2 + ^7r, leading to the same result as in Art. 76 (12). In the following table, taken (with a slight modification) from Bobyleff' s paper, the second column gives the ratio PjPo of the resultant pressure to that experienced by a plane strip of the same area. This ratio is a maximum when a = 100°, about, the lamina being then concave on the up-stream side. In the third column the ratio of P to the distance (2b sin a) between the edges of the lamina is compared with \pq^. For values of a nearly equal to 180°, this ratio tends to the value unity, as we should expect, since the fluid within the acute angle is then nearly at rest, and the pressure-excess therefore practically equal to \pq^- The last column gives the ratio of the resultant pressure to that experienced by a plane strip of breadth 26 sin a, as calculated from (12). a PlPo P/pq Q 2 b sin a PIP sin a 10° •039 •199 •227 20° •140 •359 •409 30° •278 •489 •555 40° •433 •593 •674 45° •512 •637 •724 50° •589 •677 •769 60° •733 •745 •846 70° •854 •800 •909 80° •945 •844 •959 90° 1-000 •879 1-000 100° 1-016 •907 1-031 110° •995 •931 1-059 120° •935 •950 1-079 130° •840 •964 1-096 135° •780 •970 1-103 140° •713 •975 1-109 150° •559 •984 1-119 160° •385 •990 1-126 170° •197 •996 1-132 Discontinuous Motions. 79. It must suffice to have given a few of the more important examples of steady motion with a free surface, treated by what is perhaps the most system- atic method. Considerable additions to the subject have been made by Michell*, Lovef, and other writers J. It remains to say something of the physical * "On the Theory of Free Stream-lines," Phil. Trans. A, clxxxi. (1890). + " On the Theory of Discontinuous Fluid Motions in Two Dimensions," Proc. Camb. Phil. Soc. vii. (1891). J For references see Love, Encycl. d. math. Wiss. iv. (3), 97.... A very complete account of the more important known solutions, with fresh additions and developments, is given by Greenhill, 106 Motion of a Liquid in Two Dimensions [chap, iv considerations which led in the first instance to the investigation of such problems. We have, in the preceding pages, had several instances of the flow of a liquid round a sharp projecting edge, and it appeared in each case that the velocity there was infinite. This is indeed a necessary consequence of the assumed irrotational character of the motion, whether the fluid be incom- pressible or not, as may be seen by considering the configuration of the equi- potential surfaces (which meet the boundary at right angles) in the immediate neighbourhood. The occurrence of infinite values of the velocity maybe afforded by supposing the edge to be slightly rounded, but even then the velocity near the edge will much exceed that which obtains at a distance great in comparison with the radius of curvature. In order that the motion of a fluid may conform to such conditions, it is necessary that the pressure at a distance should greatly exceed that at the edge. This excess of pressure is demanded by the inertia of the fluid, which cannot be guided round a sharp curve, in opposition to centrifugal force, except by a distribution of pressure increasing with a very rapid gradient outwards. Report on the Theory of a Stream-line past a Plane Barrier, published by the Advisory Committee for Aeronautics, 1910. The extension to the case of curved rigid boundaries is discussed in a general manner in various papers by Levi-Civita and Cisotti. For these, reference may be made to the Rend. d. Circolo Mat. di Palermo, xxiii. xxv. xxvi. xxviii. and the Rend. d. r. Accad. d. Lincei, xx. xxi. ; the working out of particular cases naturally presents great difficulties. The matter was treated later by Leathern, Phil. Trans. A, ccxx. 439 (1915) and H. Levy, Proc. Roy. Soc. A, xcii. 107 (1915). The theory of mutually impinging jets is treated very fully by Cisotti, " Vene confluenti," Ann. di mat. (3) xxiii. 285 (1914). 79] Discontinuous Motions 107 Hence, unless the pressure at a distance be very great, the maintenance of the motion in question would require a negative pressure at the corner, such as fluids under ordinary conditions are unable to sustain. To put the matter in as definite a form as possible, let us imagine the following case. Let us suppose that a straight tube, whose length is large compared with the diameter, is fixed in the middle of a large closed vessel filled with frictionless liquid, and that this tube contains, at a distance from the ends, a sliding plug, or piston, P, which can be moved in any required manner by extraneous forces applied to it. The thickness of the walls of the tube is supposed to be small in comparison with the diameter; and the edges, at the two ends, to be rounded off, so that there are no sharp angles. Let us further suppose that at some point of the walls of the vessel there is a lateral tube, with a piston P, by means of which the pressure in the interior can be adjusted at will. Everything being at rest to begin with, let a slowly increasing velocity be communicated to the plug P, so that (for simplicity) the motion at any instant may be regarded as approximately steady. At first, provided a sufficient force be applied to Q, a continuous motion of the kind indicated in the diagram on p. 74 will be produced in the fluid, there being in fact only one type of motion consistent with the conditions of the question. As the acceleration of the piston P proceeds, the pressure on Q may become enormous, even with very moderate velocities of P, and if Q be allowed to yield, an annular cavity will be formed at each end of the tube. It is not easy to make out the further course of the motion in such a case from a theoretical standpoint, even in the case of a 'perfect' fluid. In actual liquids the problem is modified by viscosity, which prevents any slipping of the fluid immediately in contact with the tube, and must further exercise a considerable influence on such rapid different motions of the fluid as are here in question. As a matter of observation, the motions of fluids are often found to differ widely, under the circumstances supposed in each case, from the types represented on such diagrams as those of pp. 73, 74, 84, 86. In such a case as we have just described, the fluid issuing from the mouth of the tube does not immediately spread out in all directions, but forms, at all events for some distance, a more or less compact stream, bounded on all sides by fluid nearly at rest. A familiar instance is the smoke-laden stream of gas issuing from a chimney. In all such cases, however, the motion in the immediate neighbour- hood of the boundary of the stream is found to be wildly irregular*. It was the endeavour to construct types of steady motion of a frictionless * Certain experiments would indicate that jets may be formed before the ' limiting velocity ' of Helmholtz is reached, and that viscosity plays an essential part in the process. Smoluchowski, " Sur la formation des veines d'efnux dans les liquides," Bull, de V Acad, de Cracovie, 1904. 108 Motion of a Liquid in Two Dimensions [chap, iv liquid, in two dimensions, which should resemble more closely what is observed in such cases as we have referred to, that led Helmholtz* and Kirchhoff* to investigate the theory of free stream-lines. It is obvious that we may imagine the space beyond a free boundary to be occupied, if we choose, by liquid of the same density at rest, since the condition of constant pressure along the stream-line is not thereby affected. In this way the problems of Arts. 76, 77, for example, give us a theory of the pressure exerted on a fixed lamina by a stream flowing past it, or (what comes to the same thing) the resistance experienced by a lamina when made to move with constant velocity through a liquid which would otherwise be at rest. The question as to the practical validity of this theory will be referred to later in connection with some related problems (Chapter XI.). Flow in a Curved Stratum. 80. The theory developed in Arts. 59, 60 may be readily extended to the two-dimensional motion of a curved stratum of liquid, whose thickness is small compared with the radii of curvature. This question has been discussed from the point of view of electric conduction, by Boltzmannf, KirchhoffJ, Topler§, and others. As in Art. 59, we take a fixed point A, and a variable point P, on the surface defining the form of the stratum, and denote by \jr the flux across any curve AP drawn on this surface. Then \js is a function of the position of P, and by displacing P in any direction through a small distance 8s, we find that the flux across the element 8s is given by dyjs/ds . 8s. The velocity perpendicular to this element will be 8^jh8s, where h is the thick- ness of the stratum, not assumed as yet to be uniform. If, further, the motion be irrotational, we shall have in addition a velocity -potential 0, and the equipotential curves = const, will cut the stream-lines x// = const, at right angles. In the case of uniform thickness, to which we now proceed, it is convenient to write yjs for y\rjh, so that the velocity perpendicular to an element 8s is now given indifferently by d\]//ds and dcfy/dn, 8n being an element drawn at right angles to 8s in the proper direction. The further relations are then exactly as in the plane problem ; in particular the curves = const., \f/ = const., drawn for a series of values in arithmetic progression, the common difference being infinitely small and the same in each case, will divide the surface into elementary squares. For, by the orthogonal property, the elementary spaces in question are rectangles, and if 8s lf 8s 2 be elements of a stream-line and an equipotential line, respectively, forming the sides of one of these rectangles, we have d^/ds 2 = d(f>/ds 1 , whence 8s l = 8s 2 , since by construction 8\f/ = 8<p. Any problem of irrotational motion in a curved stratum (of uniform thickness) is therefore reduced by orthomorphic projection to the corresponding problem in piano. Thus for a spherical surface we may use, among an infinity of other methods, that of stereographic projection. As a simple example of this, we may take the case of a stratum * 11. c. ante pp. 75, 94. t Wiener Sitzungsberichte, lii. 214 (1865) [Wissenschaftliche Abhandlungen, Leipzig, 1909, i. 1]. J Berl. Monatsber. July 19, 1875 [Ges. Abh. i. 56]. § Pogg. Ann. clx. 375 (1877). 79-8o] Flow in a Curved Stratum 109 of uniform depth covering the surface of a sphere with the exception of two circular islands (which may be of any size and in any relative position). It is evident that the only (two-dimensional) irrotational motion which can take place in the doubly-connected space occupied by the fluid is one in which the fluid circulates in opposite directions round the two islands, the cyclic constants being equal in magnitude. Since circles project into circles, the plane problem is that solved in Art. 64, 2°, viz. the stream-lines are a system of coaxal circles with real 'limiting points' (A, B, say), and the equipotential lines are the orthogonal system passing through A, B. Returning to the sphere-, it follows from well-known theorems of stereographic projection that the stream-lines (including the contours of the two islands) are the circles in which the surface is cut by a system of planes passing through a fixed line, viz. the intersection of the tangent planes at the points corresponding to A and B, whilst the equipotential lines are the circles in which the sphere is cut by planes passing through these points*. In any case of transformation by orthomorphic projection, whether the motion be irrotational or not, the velocity (d\j//dn) is transformed in the inverse ratio of a linear element, and therefore the kinetic energies of the portions of the fluid occupying corre- sponding areas are equal (provided, of course, the density and the thickness be the same). In the same way the circulation (jd\J//dn.ds) in any circuit is unaltered by projection. * This example was given by Kirchhoff, in the electrical interpretation, the problem considered being the distribution of current in a uniform spherical conducting sheet, the electrodes being situate at any two points A, B of the surface. CHAPTER V 1RR0TATI0NAL MOTION OF A LIQUID: PROBLEMS IN THREE DIMENSIONS 81. Of the methods available for obtaining solutions of the equation V 2 <£ = (1) in three dimensions, the most important is that of Spherical Harmonics. This is especially suitable when the boundary conditions have relation to spherical or nearly spherical surfaces. For a full account of this method we must refer to the special treatises*, but as the subject is very extensive, and has been treated from different points of view, it may be worth while to give a slight sketch, without formal proofs, or with mere indications of proofs, of such parts of it as are most important for our present purpose. It is easily seen that since the operator V 2 is homogeneous with respect to x, y, z, the part of <f> which is of any specified algebraic degree must satisfy (1) separately. Any such homogeneous solution of (I) is called a 'spherical solid harmonic' of the algebraic degree in question. If cf) n be a spherical solid harmonic of degree n, then if we write <pn = r"S n , (2) S n will be a function of the direction (only) in which the point (x, y, z) lies with respect to the origin; in other words, a function of the position of the point in which the radius vector meets a unit sphere described with the origin as centre. It is therefore called a 'spherical surface harmonic' of order n\. To any solid harmonic $ n of degree n corresponds another of degree — n — 1, obtained by division by r 2n+1 ; i.e. (j> = r~ 2/l_1 </> n is also a solution of (1). Thus, corresponding to any spherical surface-harmonic S n , we have the two spherical solid harmonics r n S n and r~ n ~ 1 S n . 82. The most important case is when n is integral, and when the surface - harmonic S n is further restricted to be finite over the unit sphere. In the * Todhunter, Functions of Laplace, Lame, and Bessel, Cambridge, 1875. Ferrers, Spherical Harmonics, Cambridge, 1877. Heine, Handbuch der Kugelfunctionen, 2nd ed., Berlin, 1878. Thomson and Tait, Natural Philosophy, 2nd ed., Cambridge, 1879, i. 171-218. Byerly, Fourier's Series and Spherical, Cylindrical, and Ellipsoidal Harmonics, Boston, U.S.A. 1893. Whittaker and Watson, Modern Analysis, 3rd ed. , Cambridge, 1920. For the history of the subject see Todhunter, History of the Theories of Attraction, dx., Cambridge, 1873, ii. Also Wangerin, " Theorie d. Kugelfunktionen, u.s.w.," Encycl. d. math. Wiss. ii. (1) (1904). t The symmetrical treatment of spherical solid harmonics in terms of Cartesian co-ordinates was introduced by Clebsch, in a much neglected paper, Crelle, lxi. 195 (1863). It was adopted independently by Thomson and Tait as the basis of their exposition. 81-82] Spherical Harmonics 111 form in which the theory (for this case) is presented by Thomson and Tait, and by Maxwell*, the primary solution of (1) is 4>-i = A/r (3) This represents as we have seen (Art. 56) the velocity potential due to a point-source at the origin. Since (1) is still satisfied when <f> is differ- entiated with respect to x, y, or z, we derive a solution ^- A \ l to+ m fr+ n dik - (4) dy This is the velocity-potential of a double source at the origin, having its axis in the direction (/, m, n); see Art. 56 (3). The process can be continued, and the general type of spherical solid harmonic obtainable in this way is d 11 1 *-»- 1= ^ dh 1 dh 2 ...dh M r' (5) , 9. . a 3 a where aT s = ^ +m ^ +,!s fe' l s , m s , n s being arbitrary direction-cosines. This may be regarded as the velocity-potential of a certain configuration of simple sources about the origin, the dimensions of this system being small compared with r. To construct this system we premise that from any given system of sources we may derive a system of higher order by first displacing it through a space ^7* s in the direction (l s , m s , n s ), and then superposing the reversed system, supposed displaced from its original position through a space \h s in the opposite direction. Thus, beginning with the case of a simple source at the origin, a first application of the above process gives us two sources 0+, 0_ equidistant from the origin, in opposite directions. The same process applied to the system + , 0_ gives us four sources + + , 0_ + , + _, 0__ at the corners of a parallelogram. The next step gives us eight sources at the corners of a parallelepiped, and so on. The velocity-potential, at a great distance, due to an arrangement of 2 n sources obtained in this way, will be given by (5), if 4nrA = m'hxh^ ... h n , m' being the strength of the original source at 0. The formula becomes exact, for all distances r, when h 1} h 2 , ... h n are diminished, and ra' increased, indefinitely, but so that A is finite. The surface-harmonic corresponding to (5) is given by ^=^" +1 9 wctJ' (6) and the complementary solid harmonic by cf> n = r"S n = r^cl>_ n _ 1 (7) * Electricity and Magnetism, c. ix. 112 Irrotational Motion of a Liquid [chap, v By the method of 'inversion*,' applied to the above configuration of sources, it may be shewn that the solid harmonic (7) of positive degree n may be regarded as the velocity-potential due to a certain arrangement of 2 n simple sources at infinity. The lines drawn from the origin in the various directions (l s , m S} n 8 ) are called the 'axes' of the solid harmonic (5) or (7), and the points in which these lines meet the unit sphere are called the 'poles' of the surface-harmonic S n . The formula (5) involves 2n + 1 arbitrary constants, viz. the angular co-ordinates (two for each) of the n poles, and the factor A, It can be shewn that this expression is equivalent to the most general form of spherical surface-harmonic which is of integral order n and finite over the unit sphere f. 83. In the original investigation of Laplace f, the equation V 2 = O is first expressed in terms of spherical polar co-ordinates, r, 0, co, where x = r cos 0, y = r sin cos co, z = r sin sin co. The simplest way of effecting the transformation is to apply the theorem of Art. 36 (2) to the surface of a volume-element rS0 .rsin 08co . 8r. Thus the difference of flux across the two faces perpendicular to r is ~-( ^- . r80 . r sin 08co\ 8r. Similarly for the two faces perpendicular to the meridian (co — const.) we find U%e- rsi » es '°- Sr ) Sd > and for the two faces perpendicular to a parallel of latitude (0 = const.) oco \r sin uoco } Hence, by addition, *"s(-59*S(*'»*=»3-* « This might of course have been derived from Art. 81 (1) by the usual method of change of independent variables. If we now assume that cj> is homogeneous, of degree n, and put we obtain J- ^(sin 6 ^ +-\^^ + n(n + l)S n = 0, (2) sin 000 \ 00 } sin 2 dco 2 which is the general differential equation of spherical surface-harmonics. * Explained by Thomson and Tait, Natural Philosophy, Art. 515. t Sylvester, Phil. Mag. (5), ii. 291 (1876) [Mathematical Papers, Cambridge, 1904..., iii. 37]. J "Theorie de l'attraction des sph^roides et de la figure des planetes," Mem. de VAcad. roy. des Sciences, 1782 [Oeuvres Computes, Paris, 1878... , x. 341]; Mecanique Celeste, Livre 2 me , c. ii. 82-84] Spherical Harmonics 113 Since the product n (n + 1) is unchanged in value when we write — n — 1 for n, it appears that will also be a solution of (1), as already stated (Art. 81). 84. In the case of symmetry about the axis of x, the term d 2 S n /dco 2 dis- appears, and putting cos 6 = fi we get l^-^fH^ 1 ^ ' w the differential equation of spherical 'zonal' harmonics*. This equation, con- taining only terms of two different dimensions in /x, is adapted for integration by series. We thus obtain ( n(n+l) (n-2)n(n+l)(n + 3) &„ = ^ljl- 12 /*+ ! 2.3.1 ** "- j (n-l)(n + 2) , , (n-3)(«-l)(n + 2)(n+ i» 6 j + B T 1.2.3 ^ + 1.2.3.4.5 -/*--}■ (2) The series which here present themselves are of the kind called 'hyper- geometric'; viz. if we write, after Gauss f, F(a, 8,y,x) = l + -=— - a? + -, — z — tf 2 v ^ /y 1.7 I.2.7.7 + I c.g + l.g + 2.fl.£ + l.fl + 2 + I.2.3.7.7+I.7 + 2 * ■'•' (3) we have S„ = 41* (- \n, i + £«, i & + B/J.F (| - £», 1 + £w, f , ?) (4) The series (3) is of course essentially convergent when x lies between and 1 ; but when x=l it is convergent if, and only if, y-a-/3>0. In this case we have F(a, ft y, l). gkzlll°^|^ ), (5) where n(m) is in Gauss's notation the equivalent of Euler's r(ra + l). The degree of divergence of the series (3) when y-a-/3<0, as # approaches the value 1 , is given by the theorem J F(a, ft y, #)=(l-#)Y-*-e^( 7 -a, y-ft y, *) (6) Since the latter series will now be convergent when x= 1, we see that F(a, /3, y, #) becomes divergent as (1 -x)y~ a ~& ; more precisely, for values of x infinitely nearly equal to unity, we have '^ ^SJttizt^ *-'' " ultimately. * So called by Thomson and Tait, because the nodal lines (S n = 0) divide the unit sphere into parallel belts. t I.e. ante p. 104. X Forsyth, Differential Equations, 3rd ed., London, 1903, c. vi. 114 Irrotational Motion of a Liquid [chap, v For the critical case where y — a - /3 = 0, we may have recourse to the formula d Wt- a .. „N_^ 7 which, with (6), gives in the case supposed 7 Tx F{a, ft y, #)=-f F(a + l, + 1, y + 1, x\ (8) ^F (a, fr y, x) = ^{\ -x)~K F (y -a, y - $, y+l, x) = ^(l-^)-i.^(a,fta+3 + l,^) (9) The last factor is now convergent when x = \, so that F (a, ft y, x) is ultimately divergent as log (l-x). More precisely we have, for values of x near this limit, F ^e>" + ^= n0r&h ) ] <>sih < 10 > 85. Of the two series which occur in the general expression (Art. 84 (2)) of a zonal harmonic, the former terminates when n is an even, and the latter when n is an odd integer. For other values of n both series are essentially convergent for values of /jl between + 1, but since in each case we have y — a — /3 = 0, they diverge at the limits //, = + 1, becoming infinite as logO-V). It follows that the terminating series corresponding to integral values of n are the onhy zonal surface -harmonics which are finite over the unit sphere. If we reverse the series we find that both these cases (n even, and n odd) are included in the formula* _ 1.3.5...(2n-l) | _ n(n-l) rnW ~ 1.2. 3. ..7i r 2(2n-l)^ n{ n-l)(n-2)(n-S) ) + 2.4,(Z*-l)(2n-3) ^ ""]' V; where the constant factor has been adjusted so as to make P n (/x) = l for /u, = lf. The formula may also be written ^to-vhrS^- 1 * (2) The series (1) may otherwise be obtained by development of Art. 82 (6), which in the case of the zonal harmonic assumes the form d n 1 S n =Ar^^ n - (3) dx n r * For n even this corresponds to A = ( - Y n ' — r^ , B = ; whilst for n odd we have v ; 2 . 4 ... n A = 0, B = (-f n - 1] ~ 8 ; 5 V n it • See Heine, i. 12, 147. 2 . 4 ... (n- 1) t Tables of P 1? P 2 , ... P 7 were calculated by Glaisher, for values of fi at intervals of -01, Brit. Ass. Report, 1879, and are reprinted by Dale, Five-Figure Tables..., London, 1903. A table of the same functions for every degree of the quadrant, calculated under the direction of Prof. Perry, was published in the Phil. Mag. for Dec. 1891. Both tables are reproduced in Byerly's treatise, also by Jahnke and Emde, Funktionentafeln, Leipzig, 1909. The values of the first 20 zonal harmonics, at intervals of 5°, have been calculated by Prof. A. Lodge, Phil. Trans. A. cciii. (1904). 84-85] Zonal Harmonics 115 As particular cases of (2) we have PeO*) = l, Px 0*) = /*, P 2 W = i(3/i 2 -l), P 3 (/,) = ! (5/r> -3/4 Expansions of P n in terms of other functions of 6 as independent variables, in places of fx, have been obtained by various writers. For example, we have PJ C os^) = l-^^sin^ + (W - 1)n S: + 22 1)(W + 2) sin ^- < 4 > This may be deduced from (2)*, or it may be obtained independently by putting /jl = 1 — 2z in Art. 84 (1), and integrating by a series. The function P n (p) was first introduced into analysis by Legendre t as the coefficient of k n in the expansion of The connection of this with our present point of view is that if cf> be the velocity-potential of a unit source on the axis of x at a distance c from the origin, we have, on Legendre's definition, for values of r less than c, 47r<£ = (c 2 - 2/icr + r 2 )~h T l* 2 -£+*?+*?+ < B > Each term in this expansion must separately satisfy V 2 (£ = 0, and therefore the coefficient P n must be a solution of Art. 84 (1). Since P n , as thus defined, is obviously finite for all values of /x, and becomes equal to unity for /x= 1, it must be identical with (1). For values of r greater than c, the corresponding expansion is ^=l+ p 4 +p 4 + (6) We can hence deduce expressions, which will be useful to us later, Art. 98, for the velocity-potential due to a double-source of unit strength, situate on the axis of x at a distance c from the origin, and having its axis pointing from the origin. This is evidently equal to dcf)/dc, where $ has either of the above forms; so that the required potential is, for r<c, -l&* p *? +3P 4- ■■■) < 7 > andforr> c , s{ Pl h +iP ^*-) (8) The remaining solution of Art. 84 (1), in the case of n integral, can be put into the more compact form} e»o*)=iP»(^)iog^-^, (9) where *_-_. P^ + — ^ P„_, + (10) * Murphy, Elementary Principles of the Theories of Electricity, <£c, Cambridge, 1833, p. 7. [Thomson and Tait, Art. 782.] t "Sur l'attraction des spheroides homogenes," Mem, des Savans Etrangers, x. (1785). % This is equivalent to Art. 84 (4) with, for n even, A = 0, B=(-)$ n : — m " n ; whilst for n odd we have^ = (-)* (w+1) 2 - a 4 -:- (n X) , B = 0. See Heine, i. 141, 147. 5 . o ... n 116 Irrotational Motion of a Liquid [chap, v This function Q n ^) is sometimes called the zonal harmonic 'of the second kind.' Thus Qo M - i log \±£, Q, W = 1 (3/, 2 - 1) log \±t - ifli A. — fJL JL — fJU 86. When we abandon the restriction as to symmetry about the axis of x, we may suppose S n , if a finite and single- valued function of o>, to be expanded in a series of terms varying as cos sw and sin sco respectively. If this expansion is to apply to the whole sphere (i.e. from co = to co = 2tt), we may further (by Fourier's theorem) suppose the values of s to be integral. The differential equation satisfied by any such term is |^-^f} + H» +i )-^} s »= o « If we put S n = (1 - fi 2 )^ s v, this takes the form (i_ ' t2) S" 2(s+i)M £ +( "" s)(n+s+i)t,=0, which is suitable for integration by series. We thus obtain t (n - s - 2) (n - s) (n + s + 1) (n -f s + 3) 4 ) + 1.2.3.4 A*--v| , (n - g - 3) (n - j - l)(n + g + 2) (n + j + 4) " B ) ,~ 1.2.3.4.5 the factor cos sco or sin s&> being for the moment omitted. In the hyper- geometric notation this may be written Sn-il-^lAF^s-in^ + is + in,^^) + BfMF(i + is-in,l+liS + in,§,^)}. ...(3) These expressions converge when /a 2 < 1, but since in each case we have y — a — /9 = -5, the series become infinite as (1 - /j?)~ s at the limits /n = + 1, unless they terminate*. The former series terminates when n — s is an even, and the * Kayleigh, Theory of Sound, London, 1877, Art. 338. 85-86] Tesseral Harmonics 117 latter when it is an odd integer. By reversing the series we can express both these finite solutions by the single formula* P s (a) _ ( 2n ) 1 a _ ^h Ln- S _ {n-s){n-s-l) 2 rn W -2"(n-s)\n\ {i M r 2.(2n-l) ^ (ft-s)(n-s-l)(ft-s-2)(?i-s-3) _ 4 _ ) , . "*"" 2.4.(2n-l)(2n-3) ^ — J-— v ; On comparison with Art. 85 (1) we find that P n 'W = (l-^*™> (5) That this is a solution of (1) may of course be verified independently. In terms of sin \ 0, we have d «/ a\ (n+s)\ • ,/jf-, (n-s)(n + s + l) . 21z) P n s (cos 0) = —, f— — sm s ^ 1 - =-^ z-r sin 2 \Q 71 v ' 2*(n-s)\s\ \ 1 .(s + 1) 2 + 1.2.(. + l)(. + 2) sin^-...|....(6) This corresponds to Art. 85 (4), from which it can easily be derived. Collecting our results we learn that a surface-harmonic which is finite over the unit sphere is necessarily of integral order, and is further expressible, if n denote the order, in the form S n = A P n (ji) + 2, (A 8 cos sm +B 8 sin SG>)P n 8 (ti), (7) containing 2n 4- 1 arbitrary constants. The terms of this involving a> are called ' tesseral ' harmonics, with the exception of the last two, which are given by the formula (1 — fj?)% n (A n cos nco + B n sin nay), and are called ' sectorial ' harmonics f ; the names being suggested by the forms of the compartments into which the unit sphere is divided by the nodal lines S n = 0. The formula for the tesseral harmonic of rank s may be obtained otherwise from the general expression (6) of Art. 82 by making n — s out of the n poles of the harmonic coincide at the point 6 = of the sphere, and distributing the remaining s poles evenly round the equatorial circle 6 = \tt. The remaining solution of (1), in the case of n integral, may be put in the form S n = ( A 8 cos sco + B s sinsco) Q n s (^), (8) * There are great varieties of notation in connection with these 'associated functions,' as they have been called. That chosen in the text was proposed by F. Neumann ; and is adopted by Whittaker and Watson, p. 323. f The prefix ' spherical ' is implied ; it is often omitted for brevity. 118 Irrotational Motion of a Liquid [chap, v where* q^^.^*^ (9 ) This is sometimes called a tesseral harmonic ' of the second kind.' 87. Two surface-harmonics S, S' are said to be ' conjugate/ or ; orthogonal/ when ffSS'dm = 0, (1) where 8vr is an element of surface of the unit sphere, and the integration ex- tends over this sphere. It may be shewn that any two surface-harmonics, of different orders, which are finite over the unit sphere, are orthogonal, and also that the 2n + 1 harmonics of any given order n, of the zonal, tesseral, and sectorial types specified in Arts. 85, 86, are all mutually orthogonal. It will appear, later, that the orthogonal property is of great importance in the physical applications of the subject. Since 8vr = sin 6808(o = — 8fju8co, we have, as particular cases of this theorem, J 1 i P m ( / ,)^ = 0, (2) J* P.W.P.00<*/**0, (3) J 1 P m s (ri.P n s(fjL)dp = 0, (4) and provided m, n are unequal. For m — n, it may be shewn f that jp^^d^^j, (5) >w=^;ra <«> 88. We may also quote the theorem that any arbitrary function f(fi, co) of the position of a point on the unit sphere can be expanded in a series of surface-harmonics, obtained by giving n all integral values from to oo , in Art. 86 (7). The formulae (5) and (6) are useful in determining the coefficients in this expansion. Thus, in the case of symmetry about an axis, the theorem takes the form /( A *) = Co+CiP 1 ( M ) + CiP 1 0*)+... + (7 n P fl (/*)+ (7) If we multiply both sides by P n (/uu) dp, and integrate between the limits ± 1, we find Ck-if f(r)dn (8) * A table of the functions Q n (/a), Q n 8 (/x), for various values of n and s, is given by Bryan, Proc. Camb. Phil. Soc. vi. 297. t Ferrers, p. 86 ; Whittaker and Watson, pp. 306, 325. 86-89] Integral Formulae 119 and, generally, C.-^£±\_ i f(t*)P.(M)dr (9) For the analytical proof of the theorem recourse must be had to the special treatises*; the physical grounds for assuming the possibility of this and other similar expansions will appear, incidentally, in connection with various problems. 89. Solutions of the equation V 2 <f> = may also be obtained by the usual method of treating linear equations with constant coefficients f. Thus, the equation is satisfied by or, more generally, by <f> = f(ax + fty 4- yz), (1) provided a 2 + /3 2 + 7 2 = (2) For example, we may put ff,ft7=l, icosS-, isin^r, (3) or, again, a, /3, 7 = 1, icoshu, sinh u (4) It may be shewn % that the most general solution possible can be obtained by superposition of solutions of the type (1). Using (3), and introducing the cylindrical co-ordinates x, m, &>, where 2/ = tfrcoso>, 2=Grsin&), (5) we build up a solution symmetrical about the axis of x if we take 1 [ 2ir <j) = ^r— f{m + VSF COS (Sr — ©)} d&. Zir J o For, since the integration extends over a whole circumference, it is immaterial where the origin of ^ is placed, and the formula may therefore be written § 1 [ 2n 1 f* <£ = ^- f (a? + im cos %)d& = - f(x + ivrcos%)d% (6) &7T J IT J This is remarkable as giving a value of <f>, symmetrical about the axis of x, in terms of its values f(x) at points of this axis. It may be shewn, by means of the theorem of Art. 38, that the form of <£ is in such a case completely determined by the values over any finite length of the axis||. As particular cases of (6) we have the functions - f * (x + %v cos *)* d&, - f " (x + t«r cos ^)- w " 1 d&, 7TJ IT J o * For an account of the more recent investigations of the question, see Wangerin, I.e. t Forsyth, Differential Equations, p. 444. % Whittaker, Month. Not. B. Ast. Soc. lxii. (1902). § Whittaker and Watson, Modern Analysis, c. xviii. II Thomson and Tait, Art. 498. 120 Irrotational Motion of a Liquid [chap, v where n will be supposed to be integral. Since these are solid harmonics finite over the unit sphere, and since, for vr = 0, they reduce to r n and r~ n ~\ they must be equivalent to P n (/jl) r n , and P n (fj) r _n_1 , respectively. We thus obtain the forms iY0") = - [*{** + *V(1 -^ 2 )cos^^, (7) 7T Jo P n (?) = - J Q ^ + i ^/ (1 _ ^ cog c^jn+l • ( 8 ) due originally to Laplace* and Jacobif, respectively. 90. As a first application of the foregoing theory let us suppose that an arbitrary distribution of impulsive pressure is applied to the surface of a spherical mass of fluid initially at rest. This is equivalent to prescribing an arbitrary value of <£ over the surface ; the value of </> in the interior is thence determinate, by Art. 40. To find it, we may suppose the given surface-value to be expanded, in accordance with the theorem quoted in Art. 88, in a series of surface-harmonics of integral order, thus <l> = So + S 1 + S t + ...+S n + (1) The required value is then for this satisfies V 2 <£ = 0, and assumes the prescribed form (1) when r— a, the radius of the sphere. The corresponding solution for the case of a prescribed value of <£ over the surface of a spherical cavity in an infinite mass of liquid initially at rest is evidently a ~ a 2 „ a 3 a n+1 ^ <*>—«»+ J3&+ - 3 « 2 + ... + ^s+i«»+ (3) Combining these two results we get the case of an infinite mass of fluid whose continuity is interrupted by an infinitely thin vacuous stratum, of spherical form, within which an arbitrary impulsive pressure is applied. The values (2) and (3) of <f> are of course continuous at the stratum, but the values of the normal velocity are discontinuous, viz. we have, for the internal fluid, or a and for the external fluid §* __:£(„ + 1)* or 'a * Mec. Cel. Livre ll me , c. ii. t Crelle, xxvi. (1843) [Gesammelte Werke, Berlin, 1881... , vi. 148]. 89-9i] Applications 121 The motion, whether internal or external, is therefore that due to a distribution of simple sources with surface-density 2(2n+l)^ (4) over the sphere ; see Art. 58. 91. Let us next suppose that, instead of the impulsive pressure, it is the normal velocity which is prescribed over the spherical surface ; thus d £ = S 1 + S i +...+S n +..., (1) the term of zero order being necessarily absent, since we must have d<f> w ,,.^ = 0, (2) on account of the constancy of volume of the included mass. The value of <f> for the internal space is of the form <f> = A 1 rS 1 + A 2 r 2 S 2 +...+A n r n S n + ..., (3) for this is finite and continuous, and satisfies V 2 <£ = 0, and the constants can be determined so as to make d<f>/dr assume the given surface- value (1); viz. we have 7iJ. n a n_1 = 1. The required solution is therefore 1 r n <t> = a2±^-S n (4) Y na n n v ' The corresponding solution for the external space is found in like manner to be *=- a% l^% 8 » (5) The two solutions, taken together, give the motion produced in an infinite mass of liquid which is divided into two portions by a thin spherical membrane, when a prescribed normal velocity is given to every point of the membrane, subject to the condition (2). The value of <f> changes from aXS n /n to — aXS n /(n + 1), as we cross the membrane, so that the tangential velocity is now discontinuous. The motion, whether inside or outside, is that due to a double-sheet of density -» 2 ^^ w see Art. 58. The kinetic energy of the internal fluid is given by the formula (4) of Art. 44, viz. 2T=pjJ4> d ^dS^pa^ljjS^d^ (7) the parts of the integral which involve products of surface-harmonics of different orders disappearing in virtue of the orthogonal property of Art. 87. 122 Irrotational Motion of a Liquid [chap, v For the external fluid we have *T — p\\*%iS-par2.^ rx \\lUdm (8) 91 a. The harmonic of zero order lends itself at once to the discussion of the two mathematically cognate problems of the collapse of a spherical bubble yi water, and the expansion of a spherical cavity due to the pressure of an included gas, as in the case of a submarine mine. In the former problem*, if R Q be the initial radius of the bubble, and R its value at time t, we have *-*?, a) since this makes — dcp/dr=R, for r=R. Hence, putting G = in Art. 22 (5), we have p-po _ R 2 R + 2RR* _ RtR 2 m p ~ r 2r* ' { } if p be the pressure at r=oo . Hence, putting r — R and neglecting the internal pressure RR+%&=-?-\ (3) the integral of which is jfil&=%& (Ro 3 -R 3 ) (4) This cannot easily be integrated further, but the time fa) of total collapse can be found ; thus, putting R = R x^ y ^^N/(4)/o*"* (1 -* r4 ^=^V(^ I ^ )= " 916 * ,V0>w --" t5) Thus if p = l, Ro=I cm., and ^ =l° 6 C.G.s. (1 atmosphere), ^ = -000915 sec. The kinetic energy at any instant is 27rpR?R 2 = §np (R 3-R 3 ), (6) as is indeed obvious from a consideration of the work done at a distance on the fluid. When, the collapse occurs, the energy destroyed, or rather converted into other forms, is inpoRo 3 . If ^0 = 1, Po= 10 6 , this is 4'18 x 10 6 ergs, or about -308 of a ft. -lb. The equations (1) and (2) are applicable also to the problem of the expanding cavity, but we now negleut the pressure p at a distance. If p t be the initial pressure in the cavity, when R = R , and i2=0, the internal pressure at time t is given by ,H§f <* if we assume the adiabatic law of expansion. Hence RR+%R2 = c *(^\ (8) where c = \/(M>) (9) This quantity c is of the nature of a velocity, and determines the rapidity with which changes take place. The integral of (8) is $-,-^m-(m <■»> * Besant, Hydrostatics and Hydrodynamics, Cambridge, 1859; Kayleigh, Phil. Mag. xxxiv. 94 (1917) [Papers, vi. 504]. 91-92] Radial Motion 123 It appears from (8) that the initial acceleration (R) in the radius is c 2 /i2 , whatever the law of expansion. For (8) and (10) we find that the maximum of i? occurs when (RJR fy-* = y, (11) and is given by — s = ; — 77 (12) The solution is not easily completed except in the special case of y = $. Writing R/fio=l+z, (13) we have then (i +2 )2^ = ^ // 2 A (14) ■ - whence c */22 = > /(2;s)(l+§z+^ 2 ). (15) As a concrete illustration, suppose the initial diameter of the cavity to be 1 metre, and the initial pressure pi to be 1000 atmospheres, which makes c = 3'16x 10 4 cm. /sec. It is then found that the radius of the cavity is doubled in ^\q of a second, and multiplied five-fold in about -fa sec. The initial acceleration of the radius is 2'OOx 10 7 cm./sec. 2 , shewing that the neglect of gravity in the early stages of the motion is amply justified. The maximum of R occurs when R/R = $, £='0016 sec, and is about 145 metres per second, or about one-tenth of the velocity of sound in water. With initial pressures of the order of 10,000 atmospheres or more, we should have velocities comparable with the velocity of sound, and the effect of compressibility would be no longer negligible*. 92. The harmonic of the first order is involved in the problem of the motion of a solid sphere in an infinite mass of liquid which is at rest at infinity. If we take the origin at the centre of the sphere, and the axis of x in the direction of motion, the normal velocity at the surface is Ux/r, = £7" cos 0, where U is the velocity of the centre. Hence the conditions to determine <£ are (1°) that we must have V 2 <£ = everywhere, (2°) that the space-derivatives of <t> must vanish at infinity, and (3°) that at the surface of the sphere (r = a) we must have - d ^=Ucosd (1) or The form of this suggests at once the zonal harmonic of the first order; we therefore assume ,81 . cos 6 ox r r 1 The condition (1) gives — 2A/a?= U, so that the required solution isf = £ET^cos0 (2) It appears on comparison with Art. 56 (4) that the motion of the fluid is the same as would be produced by a double-source of strength 27r Ua?, situate at the centre of the sphere. For the forms of the lines of motion see p. 128. * This discussion is taken from a paper "The early stages of a submarine explosion," Phil. Mag. xlv. 257 (1923). t Stokes, "On some cases of Fluid Motion," Camb. Trans, viii. (1843) [Paper?, i. 17]. Dirichlet, "Ueber die Bewegung eines festen Korpers in einem incompressibeln fliissigen Medium," Berl. Monatsber. 1852 [Werke, Berlin, 1889-97, ii. 115]. 124 Irrotational Motion of a Liquid [chap, v To find the energy of the fluid motion we have 2T=- P (U d ^dS = $ P aU 2 rcos 2 d.27rasm0.add = f7r /3 a 3 /7 2 = i/ / ^ 2 , (3) if M' — \ irpa z . It appears, exactly as in Art. 68, that the effect of the fluid pressure is equivalent simply to an addition to the inertia of the solid, the increment being now half the mass of the fluid displaced *. Thus in the case of rectilinear motion of the sphere, if no external forces act on the fluid, the resultant pressure is equivalent to a force - M,d £> <*> in the direction of motion, vanishing when U is constant. Hence if the sphere be set in motion and left to itself, it will continue to move in a straight line with constant velocity. The behaviour of a solid projected in an actual fluid is of course quite different; a continual application of force is necessary to maintain the motion, and if this be not supplied the solid is gradually brought to rest. It must be remembered, however, in making this comparison, that in a 'perfect' fluid there is no dissipation of energy, and that if, further, the fluid be incompressible, the solid cannot lose its kinetic energy by transfer to the fluid, since, as we have seen in Chapter ill., the motion of the fluid is entirely determined by that of the solid, and therefore ceases with it. If we wish to verify the preceding result by direct calculation from the formula ;-¥-**+*(* m we must remember that the origin is in motion, and that the values of r and 6 for a fixed point of space are therefore increasing at the rates — £7 cos 0, and (ZJsin 6)/r, respectively ; or we may appeal to Art. 20 (6). In either way we find P = $a^COs6 + 1 \U2cOS20-^U* + F(t) ( 6 ) The last three terms are the same for surface-elements in the positions Q and it — 6 ; so that, when U is constant, the pressures on the various elements of the anterior half of the sphere are balanced by equal pressures on the corresponding elements of the posterior half. But when the motion of the sphere is being accelerated there is an excess of pressure on the anterior, and a defect on the posterior half. The reverse holds when the motion is being retarded. The resultant effect in the direction of motion is dU as before. ■/, 27ra sin 6 . ad6 . p cos &= — %7rpa 3 _ 93. The same method can be applied to find the motion produced in a liquid contained between a solid sphere and a fixed concentric spherical boundary, when the sphere is moving with given velocity U. * Stokes, I.e. The result had been obtained otherwise, on the hypothesis of infinitely small motion, by Green, "On the Vibration of Pendulums in Fluid Media," Edin. Trans. 1833 [Papers, p. 315]. 92-94] Motion of a Sphere 125 The centre of the sphere being taken as origin, it is evident, since the space occupied by the fluid is limited both externally and internally, that solid harmonics of both positive and negative degrees are admissible ; they are in fact required, in order to satisfy the boundary conditions, which are — 9<£/9r = UcosB, for r = a, the radius of the spheres, and for r=b, the radius of the external boundary, the axis of x being as before in the direction of motion. We therefore assume <£= \Ar + -\ cos B, (1) and the conditions in question give . 2B TT . 2B a 3 b 3 ' whence A= V ^-,U, B=ij^- 3 U. ,..(2) b 3 - a 3 *b 3 — a 3 The kinetic energy of the fluid motion is given by the integration extending over the inner spherical surface, since at the outer we have d<j>/dr=0. We thus find 2 ^=i"^^V^ 2 ( 3 ) It appears that the effective addition to the inertia of the sphere is now* 2 & 3 + 2« 3 3 fA , ^ w^ pa (4) As b diminishes from oo to a, this increases continually from %irpa 3 to oo , in accordance with Lord Kelvin's minimum theorem (Art. 45). In other words, the introduction of a rigid spherical partition in the problem of Art. 92 acts as a constraint increasing the kinetic energy for any given velocity of the sphere, and so virtually increasing the inertia of the system. 94. In all cases where the motion of a liquid takes place in a series of planes passing through a common line, and is the same in each such plane, there exists a stream-function analogous in some of its properties to the two- dimensional stream-function of the last Chapter. If in any plane through the axis of symmetry we take two points A and P, of which A is arbitrary, but fixed, while P is variable, then considering the annular surface generated by any line AP, it is plain that the flux across this surface is a function of the position of P. Denoting this function by 27r\/r, and taking the axis of x to coincide with that of symmetry, we may say that ^isa function of x and -cr, where x is the abscissa of P, and <bt, = (y 2 4- z 2 )^, is its distance from the axis. The curves yjr = const, are evidently stream-lines. If P' be a point infinitely near to P in a meridian plane, it follows from the above definition that the velocity normal to PP' is equal to 27T^.PP" * Stokes, I.e. ante p. 123. 126 Irrotational Motion of a Liquid [chap, v whence, taking PP' parallel first to tss and then to x, u = --^-, v = -^, (1) •us ors ns ox where u and v are the components of fluid velocity in the directions of x and ns respectively, the convention as to sign being similar to that of Art. 59. These kinematical relations may also be inferred from the form which the equation of continuity takes under the present circumstances. If we express that the total flux into the annular space generated by the revolution of an elementary rectangle SxBnr is zero, we find ?r- (U . 27TOT Sct) Sx + ^— (v . 277-1*7 Bx) 8vS = 0, ox x Ovs or S (fn * ) + ^ (iro) " 0i (2) which shews that wv.dx-7jsu.d7s is an exact differential. Denoting this by dty we obtain the relations (1)*. So far the motion has not been assumed to be irrotational; the condition that it should be so is dv du _ OX Ons ' which leads to g + J±_I|± = o (3) Oar 0ns 2 ns Ons The differential equation of <f> is obtained by writing d<l> dcf> ox 0ns in (2), viz. it is -X + ^ + _-^l = o (4) Ox 2 dm 2 nsdns It appears that the functions <£ and yjr are not now (as they were in Art. 62) interchangeable. They are, indeed, of different dimensions. The kinetic energy of the liquid contained in any region bounded by surfaces of revolution about the axis is given by : ,;= lU^dS ■~'K = 2irp [</>cty, (5) — ¥■. 2irnsds nsOs The stream-function for the case of symmetry about an axis was introduced in this manner by Stokes, "On the Steady Motion of Incompressible Fluids," Camb. Trans, vii. (1842) [Papers, i. 1]. Its analytical theory has been treated very fully by Sampson, "On Stokes' Current- Function," Phil. Trans. A, clxxxii. (1891). 94-95] Stokes' Stream-Function 127 8s denoting an element of the meridian section of the bounding surfaces, and the integration extending round the various parts of this section, in the proper directions. Compare Art. 61 (2). 95. In the case of a point-source at the origin whose velocity-potential is 4>=l (i) the flux through any closed curve is numerically equal to the solid angle which the curve subtends at the origin. Hence for a circle with Ox as axis, whose radius subtends an angle 6 at 0, we have, attending to the sign, 2tt^ = - 2tt (1 - cos d). Omitting the constant term we have +-;-£ • ■••*■> The solutions corresponding to any number of simple sources situate at various points of the axis of x may evidently be superposed; thus for the double-source a 1 cos . . * = -^r = ^' (3) ■ r Wr v* sin 2 B ... wehave ^ = -^ = "^ = "T _ (4) And, generally, to the zonal solid harmonic of degree — n — 1, viz. to ♦- k £i (5) corresponds * ty — A 5-^73 (6) A more general formula, applicable to harmonics of any degree, fractional or not, may be obtained as follows. Using spherical polar co-ordinates r, 0, the component velocities along r, and perpendicular to r in the plane of the meridian, are found by making the linear element PP' of Art. 94 coincide successively with rh6 and Br, respectively, viz. they are r sin 6 rd6 y rsind dr Hence in the case of irrotational motion we have JjL-,*t*, ? — «ntf|[ (8) sin Odd dr dr W Thus if $ = r n S ni (9) where S n is a zonal harmonic of order n, we have, putting p = cos 6, * Stefan, "Ueber die Kraftlinien eines um eine Axe symmetrischen Feldes," Wied. Ann. xvii. (1882). 128 Irrotational Motion of a Liquid [chap, v The latter equation gives *=;rhr n+1(1 -" 2) f' -m which must necessarily also satisfy the former; this is readily verified by means of Art. 84 (1). Thus in the case of the zonal harmonic P n , we have as corresponding values 1 dP * = *-P„<aO, + __£- f -H»(l_ # *>5g (11) 1 dP and </ > = r- w - 1 P n ( M ), ^ = - - r~ n (I - p*) "^ , (12) Tl CifjL of which the latter must be equivalent to (5) and (6). The same relations hold of course with regard to the zonal harmonic of the second kind, Q n . 96. We saw in Art. 92 that the motion produced by a solid sphere in an infinite mass of liquid may be regarded as due to a double-source at the 95-96] Stream-Lines of a Sphere 129 centre. Comparing the formulae there given with Art. 95 (4), it appears that the stream-function due to the sphere is ^ = -itf-sin 2 (1) The forms of the lines of motion corresponding to a number of equidistant values of yjr are shewn on the opposite page. The stream-lines relative to the sphere are figured in a diagram near the end of Chapter VII. Again, the stream-function due to two double-sources having their axes oppositely directed along the axis of x will be of the form *-7?— 7?-' (2) where r ly r 2 denote the distances of any point from the positions P and Q, say, of the two sources. At the stream-surface | = Owe have ri /r 2 = (A/B)K i.e. the surface is a sphere in relation to which P and Q are inverse points. If be the centre of this sphere, and a its radius, we find AIB=OP s la 3 =a*IOQ 3 (3) This sphere may be taken as a fixed boundary to the fluid on either side, and we thus obtain the motion due to a double-source (or say to an infinitely small sphere moving along Ox) in presence of a fixed spherical boundary. The disturbance of the stream-lines by the fixed sphere is that due to a double-source of the opposite sign placed at the 'inverse' point, the ratio of the strengths being given by (3)*. This fictitious double- source may be called the ' image ' of the original one. There is also a simple construction for the image of a point-source in a fixed sphere. The image of a source m at P will consist of a source m . OQ/a at the inverse point Q, together with a line of sinks extending with uniform line-density - mja from P to the centre Of. This might be deduced by integration from the preceding result, but a direct verifica- tion is simpler. It follows at once from Art. 95 (2) that the stream-function due to a line of sources of density m would be y\r = m{r- r'), (4) where r, r are the distances of the two ends of the line from the point considered. Hence the arrangement of sources just described will give, at any point R on the sphere, ^= -m. cos RPO-m. ^§cos 0QR-- (OR-QR) (5) Since QR = OR cos ORQ + OQ cos OQR, and RPO = ORQ, this reduces to \jr = — m, a constant over the sphere. For the calculation of the force on the sphere we have recourse to zonal harmonics. Referred to as origin the velocity-potential of the original source, in the neighbourhood of the sphere, is given by ,. 1 rcosB r 2 (3cos 2 0-l) * This result was given by Stokes, " On the Resistance of a Fluid to two Oscillating Spheres," Brit. Ass. Report, 1847 [Papers, i. 230]. t Hicks, I.e. infra, p. 134. See the diagram on p. 71 ante. 130 Irrotational Motion of a Liquid [chap, v The motion reflected from the sphere will be given by a 3 cos0 a 6 (3cos 2 0-l) »/»— W+ 3cM + "> (7) since this makes djdr (<f> + <f>') = 0, for r = a. The velocity at the surface will therefore be d . , ,, v 3m . „ 5ma . „ <?= "^ (< ^" f0)== 2^ Sm ^ + ^3- Sm ^ COS ^ + (8) For an approximate result we may stop the expansion at this point. The resultant force towards P is then X=-f n pcosd.27ra 2 smede = 7r P a 2 f V q 2 sm6cosede = ^^- (9) If/ be the acceleration at when the sphere is absent, viz. f=2m 2 /c 5 , we have Z = 27rpa 3 /. (10)*. 97. Rankinef employed a method similar to that of Art. 71 to discover forms of solids of revolution which will by motion parallel to their axes generate in a surrounding liquid any given type of irrotational motion symmetrical about an axis. The velocity of the solid being U, and Bs denoting an element of the meridian, the normal velocity at any point of the surface is Udtxr/ds, and that of the fluid in contact is given by — d^lixds. Equating these and integrating along the meridian, we have ^ = -|^ 2 + const (1) If in this we substitute the value of yfr due to any distribution of sources along the axis of symmetry, we obtain the equation of a family of stream- lines. If the sum of the strengths is zero, one of these lines will serve as the profile of a finite solid of revolution past which the flow takes place. In this way we may readily verify the solution already obtained for the sphere; thus, assuming T/r = ^*7 2 /r 3 , (2) we find that (1) is satisfied for r — a, provided A = -iUa? } (3) which agrees with Art. 96 (1). By a continuous distribution of sources and sinks along the axis it has been found possible to imitate forms which have empirically been found advantageous for the profiles of air-ships. The fluid pressures can in such cases be calculated, and the results compared with experiment. 98. The motion of a liquid bounded by two spherical surfaces can be found by successive approximations in certain cases. For two solid spheres moving in the line of centres the solution is greatly facilitated by the result given at the end of Art. 96, as to the 'image' of a double-source in a fixed sphere. * Prof. G. I. Taylor, Aeronautical Research Committee, R. & M. 1166 (1928). f "On the Mathematical Theory of Stream Lines, especially those with Four Foci and upwards," Phil. Trans. 1871, p. 267 (not included in the collection referred to on p. 63 ante). 96-98] Motion of Two Spheres 131 Let a, b be the radii, and c the distance between the centres A, B. Let U be the velocity of A towards B, U' that of B towards A. Also, P being any point, let AP—r, BP=r', PAB=6, PBA = 0'. The velocity-potential will be of the form U<\>+U'4>', (1) where the functions <f> and <£' are to be determined by the conditions that V 2 = O, vy=0, (2) throughout the fluid, that their space-derivatives vanish at infinity, and that or over the surface of A. whilst 30 a/ = 0, dr .(3) cos 6\ .(4) over the surface of B. It is evident that cf> is the value of the velocity-potential when A moves with unit velocity towards B, while B is at rest ; and similarly for <£'. To find <p, we remark that if B were absent the motion of the fluid would be that due to a certain double-source at A having its axis in the direction AB. The theorem of Art. 96 shews that we may satisfy the condition of zero normal velocity over the surface of B by introducing a double-source, viz. the ' image ' of that at A in the sphere B. This image is at ff l3 the inverse point of A with respect to the sphere B; its axis coincides with AB, and its strength is -/zo^/c 3 , where fi is the strength of the original source at A, viz. Mo = 27T« 3 . The resultant motion due to the two sources at A and H\ will however violate the condition to be satisfied at the surface of the sphere A, and in order to neutralize the normal velocity at this surface, due to H u we must superpose a double-source at H^ the image of H x in the sphere A. This will introduce a normal velocity at the surface of B, which may again be neutralized by adding the image of ff 2 m &i an( ^ so on « If Mi> /*2> M3> ••• De tne strengths of the successive images, and/i,/|,/|, ... their distances from A, we have / 3 = c " 6 2 /l _ 7l /l Ml & 3 MO <* M2 a% \ 6 2 M3 & 3 M4 « 3 M3 /3 3 ' c-ft M2 (c-/ 2 ) 3 ' 6 2 Jb M5 & 3 Me a 3 c-A' M4 (^.A) 3 ' M5 /5 3 '/ •(5) and so on, the laws of formation being obvious. The images continually diminish in intensity, and this very rapidly if the radius of either sphere is small compared with the shortest distance between the two surfaces. 132 Irrotational Motion of a Liquid [chap, v The formula for the kinetic energy is 2T=-p[[(U<l>+U'4>') ((7 d ^+U' d ^\dS=LU* + 2MUU' + NU' 2 , (6) provided a) where the suffixes indicate over which sphere the integration is to be effected. The equality of the two forms of M follows from Green's Theorem (Art. 44). The value of <p near the surface of A can be written down at once from the results (7) and (8) of Art. 85, viz. we have 4^ = ( M0 + /x2 + M4+...) C -^-2(^4-^3-r...)rcos^ + &c., the remaining terms, involving zonal harmonics of higher orders, being omitted, as they will disappear in the subsequent surface-integration, in virtue of the orthogonal property of Art. 87. Hence, putting d<p/dn= — cos 0, we find with the help of (5) .(8) -£=Jp(/«o + 3M2 + 3/^ + ...)«Swpa»( 1+3^ + 3 a 6 6 6 c?fi 3 *ffifi-ftfff •) (9) It appears that the inertia of the sphere A is in all cases increased by the presence of a fixed sphere B. Compare Art. 93. The value of N may be written down from symmetry, viz. it is 1 + 3^ + 3 -_g-g— — + ...), (10) where 1=C " J fs=c- fl &2 ^ "fl c-/ 2 ' fs'=c- o-/r f'- h2 f- h2 .(ii) and so on. To calculate M we require the value of <\>' near the surface of the sphere A ; this is due to double-sources /i ', m', /* 2 ', /* 3 ', ... at distances c, c-//, c-/ 2 ', c-f 3 ', ... from A, where /i '= -27T& 3 , and (p-ffi and so on. This gives, for points near the surface of A, /i' 3 ' /s' 3 ' J5 J .(12) Mo M2 4tt0' = (fii + fi 3 ' + fi 6 ' + . . . ) — — - 2 W (c-/ 2 ) 3 (c-/ 4 ) 3 + r- J*4 r cos 6 + &c. .(13) Hence M= - p J jq>' ^ dS A = P (^ + ^ + fi 6 ' + _ a 3 b 3 ( a 3 b 3 a G b 6 ) "*" -r \ l+ JFi?=3ft+fi*m-im-tv* J l (14) 98-99] Motion of Two Spheres 133 When the ratios ajc and b/c are both small we have i=§^a3(l+3^. 3 ), M=2*p^, ^=1^^(1+3^), (15) approximately* If in the preceding results we put b = a, U' = (J, the plane bisecting AB at right angles will be a plane of symmetry, and may therefore be taken as a fixed boundary to the fluid on either side. Hence, putting c = 2h, we find, for the kinetic energy of the liquid when a sphere is in motion perpendicular to a rigid plane boundary, at a distance h from it, result due to Stokes. 2T=$«pa* (l + §|-3+...) U\ (16) 99. When the spheres are moving at right angles to the line of centres the problem is more difficult ; we shall therefore content ourselves with the first steps in the approximation, referring, for a more complete treatment, to the papers cited on p. 134. Let the spheres be moving with velocities V, V in parallel directions at right angles to A, B, and let r, 0, <o and r' f 6', a> be two systems of spherical polar co-ordinates having their origins at A and B respectively, and their polar axes in the directions of the velocities V, V. The velocity-potential will be of the form V<f>+V'<t>', with the surface-conditions f r = -cos6, ^' = 0, for,=a, (1) and |£ = 0, ^=-cos0', for/ = 6 (2) If the sphere B were absent the velocity-potential due to unit velocity of A would be i a a * -s cos 6. - r l Since r cos 6 = r' cos 6\ the value of this in the neighbourhood of B will be a 3 ±- 3 r'cos6', approximately. The normal velocity at the surface of B, due to this, will be cancelled by the addition of the term x a 3 b 3 cos & which in the neighbourhood of A becomes equal to J-^rcostf, nearly. To rectify the normal velocity at the surface of J., we add the term x a 6 b 3 cos 6 *-#- "72~- Stopping at this point, and collecting our results, we have, over the surface of A, / o 3 h 3 \ <t>=\a>(l +$-jr) cos 6, (3) a 3 and at the surface of B, = |6.— cos#' (4) * To this degree of approximation the results may be more easily obtained without the use of ' images,' the procedure being similar to that of the next Art. 134 Irrotational Motion of a Liquid [chap, v Hence if we denote by P, Q, R the coefficients in the expression for the kinetic energy, viz. 2T=PV* + 2QVV' + R V'\ (5) we have *--p//*|£^-|^(l+*^-/ «--p//#8>-t^?. < 6 > The case of a sphere moving parallel to a fixed plane boundary, at a distance A, is obtained by putting b = a, V= V, c=2h, and halving the consequent value of T ; thus S2»-iirpo»(l + ftg) F» (7) This result, which was also given by Stokes, may be compared with that of Art. 98 (16)*. Cylindrical Harmonics. 100. In terms of the cylindrical co-ordinates x, vr, <o introduced in Art. 89, the equation V 2 </> = takes the form dx* + dv 2 + *rd*T~ ] ~v*dco 2 ~ V W Tnis may be obtained by direct transformation, or more simply by expressing that the total flux across the boundary of an element 8x . 8vt . vr 8co is zero, after the manner of Art. 83. In the case of symmetry about the axis of x, the equation reduces to the form (4) of Art. 94. A particular solution is then <j> = e ±kx % (-or), provided ^(•) + ^W+^W = o ( 2 ) This is the differential equation of 'Bessel's Functions' of zero order. Its complete primitive consists, of course, of the sum of two definite functions of «r, each multiplied by an arbitrary constant. That solution which is finite for «■ = is easily found in the form of an ascending series ; it is usually denoted by GJ Q (lev;), where ^o(t)=l-g + 2r^- (3) * For a fuller analytical treatment of the problem of the motion of two spheres we refer to the following papers: W. M. Hicks, "On the Motion of two Spheres in a Fluid," Phil. Trans. 1880, p. 455; E. A. Herman, "On the Motion of two Spheres in Fluid," Quart. Journ. Math. xxii. (1887) ; Basset, "On the Motion of Two Spheres in a Liquid, &c." Proc. Lond. Math. Soc. xviii. 369 (1887). See also C. Neumann, Hydrodynamische Untersuchungen, Leipzig, 1883; Basset, Hydrodynamics, Cambridge, 1888. The mutual influence of 'pulsating' spheres, i.e. of spheres which periodically change their volume, has been studied by C. A. Bjerknes, with a view to a mechanical illustration of electric and other forces. A full account of these researches is given by his son Prof. V. Bjerknes in Vorlesungen iiber hydrodynamische Fernkrafte, Leipzig, 1900-1902. The question is also treated by Hicks, Camb. Proc. iii. 276 (1879), iv. 29 (1880), and by Voigt, Gott. Nachr. 1891, p. 37. since 99-100] Cylindrical Harmonics 135 We have thus obtained solutions of V 2 <£ = of the types* £ = e ±te Ji(M (4) It is easily seen from Art. 94 (1) that the corresponding value of the stream - function is ^-Tw^/o'N (5) The formula (4) may be recognized as a particular case of Art. 89 (6); viz. it is equivalent to 0=1 r e ±k(x+ivrcos^)^ (6) Jo(0 = - f r cos(?cos^)^ = - fV C0S *c^, (7) 7T J TT J as may be verified by developing the cosine, and integrating term by term. Again, (4) may also be identified as the limiting form assumed by a spherical solid zonal harmonic when the order (n) is made infinite, provided that at the same time the distance of the origin from the point considered be made infinitely great, the two infinities being subject to a certain relation f. Thus we may take + -£P.W-(l +£)"*.<«>. (8) where we have temporarily changed the meanings of x and <xr, viz. r = a + x } vt = 2a sin \ 6 y whilst Xn (*r)=i- v 22 y -^ + — - 22 42 M — -^-•••; ( 9 > see Art. 85 (4). If we now put k = n/a, and suppose a and n to become infinite, whilst k remains finite, the symbols x and -cr will regain their former meanings, and we reproduce the formula (4) with the upper sign in the exponential. The lower sign is obtained if we start with a n+l The same procedure leads to an expression of an arbitrary function of m in terms of the Bessel's Function of zero order {. According to Art. 88, an arbitrary function of latitude on the surface of a sphere can be expanded in spherical zonal harmonics, thus F(n) = t(n + i)P n (ri ^FW)P n {p')dp' (10) * Except as to notation these solutions are to be found in Poisson, I.e. ante p. 18. + This process was indicated, without the restriction to symmetry, by Thomson and Tait, Art. 783 (1867). % The procedure appears to be due substantially to C. Neumann (1862). 136 Irrotational Motion of a Liquid [chap, v If we denote by ct the length of the chord drawn to the variable point from the pole (0 = 0) of the sphere, we have tjs = 2a sin J 0, ^8sr = — a 2 Bfju, where a is the radius, so that the formula may be written f{m)-\*(n + \)H % {w)\*f{J)P n {*)v/dm/ (11) a Jo n 1 If we now put k = - , 8k — - , r a a and finally make a infinite, we are led to the important theorem*: /(*r)= I™ J (kv)kdk \™ f(v l )J*(W)'a'dv' (12) Jo Jo 101. If in (1) we suppose </> to be expanded in a series of terms varying as cos sco or sinsa), each such term will be subject to an equation of the form S+S+ i S*-A+-o < 13 > dor 9ot 2 ta d^r n 2 This will be satisfied by <j> = e ±kx % (ct), provided rt"(»)+iX'(-) + (*-5)x(»)-o> a*) which is the differential equation of Bessel's Functions of order sf. The solution which is finite for w — may be written % (ot) = CJ S (kxs), where Js (?) = 2 s . n (J) J 1 " 2 (2* + 2) + 2 . 4 (2 5 + 2) (2* + 4) " ' * j* '" (15) The complete solution of (14) involves, in addition, a Bessel's Function 'of the second kind' with whose form we shall be concerned at a later period in our subject f. We have thus obtained solutions of the equation V 2 <£ = 0, of the types $ = e± k *J s (k>*) C HsG> (16) * For more rigorous proofs, and for the history of the theorem, see Watson, I.e. infra. t Forsyth, Art. 100; Whittaker and Watson, c. xvii. X For the further theory of the Bessel's Functions of both kinds recourse may be had to Gray and Mathews, Treatise on Bessel Functions, 2nd ed., London, 1922, and to G. N. Watson, Theory of Bessel Functions, Cambridge, 1923, where ample references are given to previous writers. An account of the subject, from the physical point of view, will be found in Eayleigh's Theory of Sound, cc. ix., xviii., with many important applications. Numerical tables of the functions J s (f ) have been constructed by Bessel and Hansen, and more recently by Meissel (Berl. Abh. 1888). These are reproduced by Gray and Mathews, and, with valuable extensions, in Watson's treatise. Abridged tables are included in the collections of Dale and of Jahnke and Emde referred to on p. 114. 100-102] Cylindrical Harmonics 137 These may also be obtained as limiting forms of the spherical solid harmonics r n t-» , x cos) a n+1 „ . , x cos) a n \rs gin j . rn+ i n \t»j gm j with the help of the expansion (6) of Art. 86*. 102. The formula (12) of Art. 100 enables us to write down expressions, which are sometimes convenient, for the value of <£ on one side of an infinite plane (x — 0) in terms of the values of <j> or dcp/dn at points of this plane, in the case of symmetry about an axis (Ox) normal to the planef. Thus if <t> = F(er), for# = 0, (1) we have, on the side x > 0, a >= I™ e - kx Jo(kvr)kdk \" F(w')J (hm')w'dm i (2) Jo Jo Again, if -^^/(^ for# = 0, (3) we have <£ = f°V te J (k*r)dk f °" f(^')J (k^ , )^'d'm' (4) Jo Jo The exponentials have been chosen so as to vanish for x = oo . Another solution of these problems has already been given in Art. 58, from equations (12) and (11) of which we derive #sS)« (5) and /: *--l\\tf <«> respectively, where r denotes distance from the element BS of the plane to the point at which the value of <f> is required. We proceed to a few applications of the general formulae (2) and (4). 1°. If, in (4), we assume /(or) to vanish for all but infinitesimal values of c;, and to become infinite for these in such a way that /(or) 27TZ(7C?G7 = ^. we obtain 4tt$= / e- kx J Q {hw)dk i (7) J o and therefore, since Jo'= —J\, 4:7r^t=-w\ e~ kx J^km) dk, (8) by Art. 100 (5). * The connection between spherical surface-harmonics and Bessel's Functions was noticed by Mehler, " Ueber die Vertheilung d. statischen Elektricitat in einem v. zwei Kugelkalotten begrenzten Korper," Crelle, lxviii. (1868). It was investigated independently by Eayleigh, "On the Eelation between the Functions of Laplace and Bessel," Proc. Lond. Math. Soc. ix. 61 (1878) [Papers, i. 338] ; see also Theory of Sound, Arts. 336, 338. There are also methods of deducing Bessel's Functions 'of the second kind' as limiting forms of the spherical harmonics Q n (a), Q n 8 (^) C ° S j- sw; for these see Heine, i. 184, 232. f The method may be extended so as to be free from this restriction. 138 Irrotational Motion of a Liquid [chap, v By comparison with the primitive expressions for a point-source at the origin (Art. 95), we infer that f" e~ 1cX J (km)dk=-, f e~ kx J 1 (km)dk= ,™ . , (9) where r=J(x 2 + m 2 ) ; these are in fact known results* 2°. Let us next suppose that sources are distributed with uniform density over the plane area contained by the circle m=a, #=0. Using the series for J , J ly or otherwise, we find ( a j Q {kw)wdvy= < ^J 1 {ka) (10) Hence t <£ = — re-xxJoi^J^ka) — , yj,= -— re-teJ^k^J^ka)^, (11) ttolj o lc ira J o lc where the constant factor has been chosen so as to make the total flux through the circle equal to unity. 3°. Again, if the density of the sources, within the same circle, vary as ll>J(a 2 -m 2 ), we have to deal with the integral J |V (to)- 7 ^^ ) = a | o i V (tosin3)sin5^ = ^, (12) where the evaluation is effected by substituting the series form of J , and treating each term separately. Hence JL 1 f *» r n s ■ 7 dk , m r fa T „ , . , dk 9 = ~ — / e- kx J (km)smka^ r , \i,= -- — / «-« J x (km) sin fca-j-, Ana J o K ZrraJ q K .(13) if the constant factor be determined by the same condition as before §. It is a known theorem of Electrostatics that the assumed law of density makes </> constant over the circular area. It may be shewn independently that J (km) smka- r =%ir i or sin -1 — , ) h W \ (14) T n . . , dk a-s/(a 2 -z3 2 ) a Ji (km) sin ka-j-= — , or — o k m m according as m<a\\. The formulae (13) therefore express the flow of a liquid through a circular aperture in a thin plane rigid wall. Another solution will be obtained in Art. 108. The corresponding problem in two dimensions was solved in Art. 66, 1°. 4°. Let us next suppose that when x=0, we have <j> = C *J(a 2 -m 2 ) for m <a, and = for m > a. We find J Jo(km)s/(a 2 -m^)mdm=a s t " J (ka sin S) sin $ cos 2 Sd$ = a 3 ^i (ka), ...(15) provided ^ l(C )^( 1 _^ + ^ 577 _...) = .^^ (16 ) Hence, by (2), <£= - cj%-» J (km) J* ( S -^ * The former is due to Lipschitz, Crelle, lvi. 189 (1859); see Watson, p. 384. The latter follows by differentiation with respect to m and integration with respect to x. t Cf . H. Weber, Crelle, lxxv. 88 ; Heine, ii. 180. X The formula (12) has been given by various writers; see Eayleigh, Papers, iii. 98; Hobson, Proc. Lond. Math. Soc. xxv. 71 (1893). § Cf. H. Weber, Crelle, lxxv. (1873) ; Heine, ii. 192. || H. Weber, Crelle, lxxv.; Watson, p. 405. See also Proc. Lond. Math. Soc. xxxiv. 282. dk (17) 102-103] Ellipsoidal Harmonics 139 This gives, for # = 0, -(jP) =C [" J (km) am ka^+Cw j"° J ' (km) sin kadk, (18) after a partial integration. The value of the former integral is given in (14), and that of the latter can be deduced from it by differentiation with respect to m. Hence -S^^-K^-^-^M' (19) according as w < a. It follows that if 0=2/*- . U, the formula (17) will relate to the motion of a thin circular disk with velocity U normal to its plane, in an infinite mass of liquid. The expression for the kinetic energy is 2T= - P fU^dS=7rpC 2 r^a 2 -™ 2 ) 2irmdm=$7r 2 pa*C\ or 2T=§pa,3U 2 (20) The effective addition to the inertia of the disk is therefore 2/?r ( = '6366) times the mass of a spherical portion of the fluid, of the same radius. For another investigation of this question, see Art. 108. Ellipsoidal Harmonics. 103. The method of Spherical Harmonics can also be adapted to the solution of the equation W=o, (i) under boundary-conditions having relation to ellipsoids of revolution*. Beginning with the case where the ellipsoids are prolate, we write x — h cos 6 cosh 77 = kfi £, y — ^ cos co, z = -or sin co, ] where sr = A?sin d sinh 77 = k(\ -,*»)*(£»- 1)*. J'" < 2 ' The surfaces f= const., /n = const, are confocal ellipsoids and hyperboloids of two sheets, respectively, the common foci being the points (+ k, 0, 0). The value of f may range from 1 to 00 , whilst fx lies between ± 1. The co-ordinates ft, £ co form an orthogonal system, and the values of the linear elements Bs^, 8s f, Sso, described by the point (%, y, z) when fi, f, co separately vary are (3) To express (1) in terms of our new variables we equate to zero the total flux across the walls of a volume element Bs^Bs^Bs^, and obtain or, on substitution from (3), * Heine, "Ueber einige Aufgaben, welche auf partielle Differentialgleichungen fuhren," Crelle, xxvi. 185 (1843), and Kugelfunctionen, ii. Art. 38. See also Ferrers, c. vi. 140 Irrotational Motion of a Liquid [chap, v This may also be written d \a ^ 8 *1 i '*■ y +-» fa m d +} ■ x 9 ^ (4) 3 M 1 (1 * } a M | + i - ff do,* ~ i% r ? } art + rrp g^-2 w 104. If <f> be a finite function of /z. and co, from /*=— lto/Lt = + l and from co = to <w = 27r, it may be expanded in a series of surface harmonics of integral orders, of the types given by Art. 86 (7), where the coefficients are functions of f ; and it appears on substitution in (4) that each term of the expansion must satisfy the equation separately. Taking first the case of the zonal harmonic, we write + -p n (r).z, (5) and on substitution we find, in virtue of Art. 84 (1), M {1 -^ d i\ +n(n+1)Z ^ ' (6) which is of the same form as the equation referred to. We thus obtain the solutions *-P,0«).P„(t), (7) and +-P»6»)-.ft.(tX ( 8 ) where C»(f)=Pn(t) ] f {p n(f )Jl(fl_l)' = ^ ! Wi . (n + l)(n + 2) n _ 3 1.3...(2n + l) ( fe 2(2^ + 3) * (»+l)(n + 2)(n + 3)(n+4) 5 1 , gv * * 2.4(2n + 3)(2n + 5) * J* " ; The solution (7) is finite when f = 1, and is therefore adapted to the space within an ellipsoid of revolution; while (8) is infinite for £= 1, but vanishes for f = oo , and is therefore appropriate to the external region. As particular cases of the formula (9) we note « a (D = i(3? 2 -i)iog|^-J-ir. The definite-integral form of Q n shews that ^.(o^p-^^e.«)— jfii (io) The expressions for the stream-function corresponding to (7) and (8) are readily found; thus, from the definition of Art. 94, * Ferrers, c. v.; Todhunter, c. vi.; Forsyth, Arts. 96-99. 103-106] Motion of an Ovary Ellipsoid 141 g = -^-l)|, Sf-*(1-*>|* d2) Thus, in the case of (7), we have ~~n(n + l) { * _1) df "d/i \ {i fl) dp wh — *-^ ) ( 1 -^^^ Ll >^r (13) The same result will follow of course from the second of equations (12). In the same way, the stream-function corresponding to (8) is +-^< 1 -^^-<« , - 1 )^ Q < 14) 105. We can apply this to the case of an ovary ellipsoid moving parallel to its axis in an infinite mass of liquid. The elliptic co-ordinates must be chosen so that the ellipsoid in question is a member of the confocal family, say that for which f = f . Comparing with Art. 103 (2) we see that if a, c be the polar and equatorial radii, and e the eccentricity of the meridian section, we must have k = ae, & = !/«, k(tf-l)$ = c. The surface-condition is given by Art. 97 (1), viz. we must have ^ = - J EW(1 -/*■)({*-!) + const., (^ for f= f - Hence putting n = l in Art. 104 (14), and introducing an arbitrary multiplier A, we have K with the condition ^=^(l-^)(?*-l){|log|±| The corresponding formula for the velocity-potential is ■(2) .(3) #-^JKlogj£|--l} (4) The kinetic energy, and thence the inertia-coefficient due to the fluid, may be readily calculated by the formula (5) of Art. 94. 106. Leaving the case of symmetry, the solutions of V 2 <f> = when <j> is a tesseral or sectorial harmonic in fju and <w are found by a similar method to be of the types 0=p„«( /t ).p M »(?)^J sa)) a) <t> = Pn°(ri.Qn>(!;)'**\s a >, (2) 142 Irrotational Motion of a Liquid [chap, v where, as in Art. 86, P n ° (fi) = (1 - fff* - "ftffi , (3) whilst (to avoid imaginaries) we write JVC?)^ 2 -!)* 8 *!^, (4) It may be shewn that enS(r)=(_)8 (^^yi PnS(r) 'J f (p,/(?)p.(r 2 -i)' (6) *_ p, ( „*p-^» (! , ( „. ( -)«|±|: ? L I m As examples we may take the case of an ovary ellipsoid moving parallel to an equatorial axis, say that of y, or rotating about this axis. 1°. In the former case, the surface-condition is d±__ v dy for £= £o> where Fis the velocity of translation, or j* — F. * & . .(1-^coso, (8) This is satisfied by putting n = 1, * = 1, in (2), viz. ^, = 4(l- M a )i(? 2 -l)*.jjlog|±j-^ I )cos6, ; (9) the constant A being given by 4 lo 4S-J^}=-^ (io > 2°. In the case of rotation about Oy, if £l y be the angular velocity, we must have 30 _ n / dx dz\ for?=r o or ^=k*n y . — 1 /.(l-^sino) (11) Putting n = 2, s = 1, in the formula (2) we find f=4 /i (l-^)i(^-l)ijfriog|±|-3-^Jsin &)> ...(12) J. being determined by comparison with (11). 107. When the ellipsoid is of the oblate or ' planetary ' form, the appropriate co-ordinates are given by x = k cos 6 sinh 77 = &//,£ ?/ = «rcosG>, ^ = -or sin co, } ,_. where «• - A; sin (9 cosh 77 = k (1 - a* 2 )*(? 2 + 1)*. J 106-107] Motion of a Planetary Ellipsoid 143 Here f may range from to oo (or, in some applications, from — oo through to 4- oo ), whilst /ub lies between ± 1. The quadrics f = const., /a = const, are planetary ellipsoids and hyperboloids of revolution of one sheet, all having the common focal circle x = 0, ct = k. As limiting forms we have the ellipsoid f = 0, which coincides with the portion of the plane x = for which «r < k, and the hyperboloid /z = coinciding with the remaining portion of this plane. With the same notation as before we find (2) and the equation of continuity becomes a^O 1 ^a^j arr 3?J (i-/* 2 )(r 2 + i)ao> 2 u ' or lr{a-^» + r^S-il(«-+l)» + ^S..^) 3yL6 ( 3/X l-/x 2 3a> 2 3f ( V5 7 afJ ' £ 2 + l3a> 2 This is of the same form as Art. 103 (4), with i% in place of f, and the like correspondence will run through the subsequent formulae. In the case of symmetry about the axis we have the solutions *-P.0»).JPto(ft W and *.«i > «G»).9»(?). (5) n(n-l)(>i-2)(n-3) ) + 2.4(2n-l)(2n-3) & )' "' W and ^(?)=F»(0j d ^ (0|2(?2+jr = " ! I^-n-i _ (* + l)(n + 2) ._ n _ 3 1.3.5.-.(2JH-l)-l i 2(2^ + 3) & , (n + l)(n+2)(n + 3)(w + 4) 2.4(2n + 3)(2n + 5) ? " ' '" \ ' (7) the latter expansion being however convergent only when f > 1 *. As before, the solution (4) is appropriate to the region included within an ellipsoid of the family f = const., and (5) to the external space. We note that p n (0 ^ - ^1> ?>l (r) = _ _i_ ( 8 ) As particular cases of the formula (7) we have g (r)=cot-i?, gi (?) = l-rcot- 1 r > ? 2 (r)=*(3r 2 +])cot- i c-K. * The reader may easily adapt the demonstrations referred to in Art. 104 to the present case. 144 Irrotational Motion of a Liquid [chap, v The formulae for the stream-function corresponding to (4) and (5) are and +-^< 1 -*>^-« i+1 >^ do 108. 1°. The simplest case of Art. 107 (5) is when n= 0, viz. </> = ^lcot- 1 ^ (1) where f is supposed to range from — oo to + oo . The formula (10) of the last Art. then assumes an indeterminate form, but we find by the method of Art. 104, ir = Akp, (2) where /m ranges from to 1. This solution represents the flow of a liquid through a circular aperture in an infinite plane wall, viz. the aperture is the portion of the plane yz for which m < k. The velocity at any point of the aperture (f =0) is _ _ 1 <ty = A ta dco ~ (p _ w 2)£ ' since, when oo — 0, kfi — (k 2 — -sr 2 )^. The velocity is therefore infinite at the edge. Compare Art. 102, 3°. 2°. Again, the motion due to a planetary ellipsoid (?=f ) moving with velocity U parallel to its axis in an infinite mass of liquid is given by ^^(Wcot- 1 ^ ^=i^(l-^ 2 )(? 2 + l){^ T -cot- 1 rf,...(3) where A = - k If + Ww ~ cot_1 ?o Denoting the polar and equatorial radii by a and c, and the eccentricity of the meridian section by e, we have In terms of these quantities A = - Uc ^\(l-e 2 )i-~ sin- 1 e\ (4) The forms of the lines of motion, for equidistant values of yjr, are shewn on the next page. Cf. Art. 71, 3°. The most interesting case is that of the circular disk, for which e = 1, and A = 2Uc/7r. The value of <f> given in (3) becomes equal to ±A/jl, or + A (1 — -g^/c 2 )^, for the two sides of the disk, and the normal velocity to ± U. Hence the formula (4) of Art. 44 gives 2T=$pc*U*, (5) as in Art. 102 (20). 107-109] Motion of a Planetary Ellipsoid 145 X' X 109. The solutions of the equation Art. 107 (3) in tesseral harmonics are *-Pn'(ri.tf(fi.*$«» 9 (1) and *-^n < 0*).9n , (f)-S«», (2) Where Pn 8 (f) = (r 2 +l)* 8 ^f|P, (3) and *.'«?)-({* + 1)**^P, K ' (n-s)rPn W-) t ift .(f)J«(fi+i) <*> These functions possess the property *•"' <*r if ?«(?)-( ) (M _ g)!?a+1 (5) We may apply these results as in Art. 108. 146 Irrotational Motion of a Liquid [chap, v 1°. For the motion of a planetary ellipsoid (f = f ) parallel to the axis of y we have n = 1, s = 1, and thence .(6) <j> = A (1 - ft (f 2 + 1)* {^ - cot- 1 fj cos o>, with the condition — = — V ~ , for f = f , F" denoting the velocity of the solid. This gives A \$rh)-***<\-- hY [ (7) In the case of the disk (f = 0), we have A = 0, as we should expect. 2°. Again, for a planetary ellipsoid rotating about the axis of y with angular velocity f\ , we have, putting n — 2, 5 = 1, ^=^(l- / a 2 )i(r 2 +l)i|3rcot- 1 r-3 + ^i- T } S ina )) (8) with the surface-condition d<j> _ n / 3d? cte at — 1J *raf~*af. = _^rii L is . nw For the circular disk (£b = 0) this gives firA^-Mly (10) At the two surfaces of the disk we have <j> = + 2A/JL (1 - yu 2 )* to *>, ^ = + kCly (1 - iff sin », and substituting in the formula 2T=-p [U^vdvdu, we obtain IT = Jf pc 5 . O y 2 (11)* 110. In questions relating to ellipsoids with three unequal axes we may employ the more general type of Ellipsoidal Harmonics, usually known by the name of 'Lame's Functions f.' Without attempting a formal account of these functions, we will investigate some solutions of the equation V 2 <£ = 0, (1) in ellipsoidal co-ordinates, which are analogous to spherical harmonics of the first and second orders, with a view to their hydrodynamical applications. * For further solutions in terms of the present co-ordinates see Nicholson, Phil. Trans. A, ccxxiv. 49 (1924). t See, for example, Ferrers, Spherical Harmonics, c. vi.; W. D. Niven, Phil. Trans. A, clxxxii. 182 (1891) and Proc. Roy. Soc. A, lxxix. 458 (1906); Poincar^, Figures d'Equilibre d'une Masse Fluide, Paris, 1902, c. vi.; Darwin, Phil. Trans. A, cxcvii. 461 (1901) [Scientific Papers, Cambridge, 1907-11, iii. 186]; Whittaker and Watson, c. xxiii. An outline of the theory is given by Wangerin, I.e. ante p. 110. 109-1 10J Motion of a Planetary Ellipsoid 147 It is convenient to prefix an investigation of the motion of a liquid con- tained in an ellipsoidal envelope, which can be treated at once by Cartesian methods. Thus, when the envelope is in motion parallel to the axis of x with velocity U, the enclosed fluid moves as a solid, and the velocity-potential is simply <£ = — Ux. Next let us suppose that the envelope is rotating about a principal axis (say that of x) with angular velocity £l x . The equation of the surface being - + ^+-=1 (2) a 2+ 6 2+ c 2 A, W the surface-condition is ~a 2 dx~ b 2 dy c 2 hz~ & ll * z + ^W We therefore assume (j> — Ayz, which is evidently a solution of (1), and obtain, on determining the constant by the condition just written, b 2 -c 2 Hence, if the centre be moving with a velocity whose components are U, V, W and if H Xi Cl y , £l z be the angular velocities about the principal axes, we have by superposition* b 2 — c 2 c 2 - a 2 a 2 — b 2 $ = - Ux - Vy - Wz --^—- 2 n x yz - -^— 2 n y zx - - z —^n z xy. ...(3) We may also include the case where the envelope is changing its form, but so as to remain ellipsoidal. If in (2) the lengths (only) of the axes are changing at the rates a, b, c, respectively, the general boundary-condition, Art. 9 (3), becomes a^+b^ + c^+atdx + Vdy + ctdz- ' W which is satisfied f by ♦-■-tg'+jii'+j*) W The equation (1) requires that & - + b 7 + ° = (6) a b c v 7 which is in fact the condition which must be satisfied by the changing ellip- soidal surface in order that the enclosed volume (%irabc) may be constant. * This result appears to have been published independently by Beltrami, Bjerknes, and Maxwell, in 1873. See Hicks, "Beport on Kecent Progress in Hydrodynamics," Brit. Ass. Rep. 1882, and Kelvin's Papers, iv. 197 (footnote). t C. A. Bjerknes, " Verallgemeinerung des Problems von den Bewegungen, welche in einer ruhenden unelastischen Fliissigkeit die Bewegung eines Ellipsoids hervorbringt," Gottinger Nachrichten, 1873, pp. 448, 829. 148 Irrotational Motion of a Liquid [chap, v 111. The solutions of the corresponding problems for an infinite mass of fluid bounded internally by an^ellipsoid involve the use of a special system of orthogonal curvilinear co-ordinates. If x, y y z be functions of three parameters X, fi, v, such that the surfaces X = const., fjL — const., v — const (1) are mutually orthogonal at their intersections, and if we write h x 2 ~ \d\J ^{dxj + \dx) ' _1 (dx\ 2 (dy\ 2 (dz\ 2 % \o"/ vW \<W •(2) the direction-cosines of the normals to the three surfaces which pass through (x, y, z) will be (^!'*!'*»i9- (^I'^l'* 8 !)' ( A '£-^l' A3 a-3'- (3) respectively. It follows that the lengths of linear elements drawn in the directions of these normals will be BX/h 1} BfM/h 2} Bv/h 3 . Hence if $ be the velocity-potential of a fluid motion, the total flux into the rectangular space included between the six surfaces X ± ^8X, //, ± -JS/*, v ± 1 8v will be It appears from Art. 42 (3) that the same flux is expressed by V 2 <£ multiplied by the volume of the space, i.e. by 8x8/jL8v/h 1 h 2 h 3 . Hence* ^-^{k(&S?) + 5(oc© + s(&S}'- - (4) Equating this to zero, we obtain the general equation of continuity in orthogonal co-ordinates, of which particular cases have already been investi- gated in Arts. 83, 103, 107. The theory of triple orthogonal systems of surfaces is very attractive mathematically, and abounds in interesting and elegant formulae. We may note that if X, fi, v be regarded as functions of x, y, z, the direction-cosines * The above method was given in a paper by W. Thomson, " On the Equations of Motion of Heat referred to Curvilinear Co-ordinates," Camb. Math. Journ. iv. 179 (1843) [Papers, i. 25]. Reference may also be made to Jacobi, "Ueber eine particulate Losung der partiellen Diffe- rentialgleichung ," Crelle, xxxvi. 113 (1847) [Werke, ii. 198]. The transformation of V 2 to general orthogonal co-ordinates was first effected by Lame, " Sur les lois de l'equilibre du fluide £th£re," Journ. de VEcole Polyt. xiv. 191 (1834). See also his Legons sur les Coordonne.es Curvilignes, Paris, 1859, p. 22. 111-112] Orthogonal Co-ordinates 149 of the three line-elements above considered can also be expressed in the forms 1 3X 1 3X 1 3X\ /l dfi I dp 1 d/i\ /l dv 1 dv 1 9i/> 9-0 V^i 3# ' h-idy' Ai 3.sv ' \A 2 3# ' h 2 dy ' h 2 dz J ' \h 3 dx ' h z dy' h z dz/ (5) from which, and from (3), various interesting relations can be inferred. The formulae already given are, however, sufficient for our present purpose. 112. In the applications to which we now proceed the triple orthogonal system consists of the confocal quadrics x 2 y 2 z 2 W+e + v + e + ?+0 -1 = 0, .(i) whose properties are explained in books on Solid Geometry. Through any given point (x, y, z) there pass three surfaces of the system, corresponding to the three roots of (1), considered as a cubic in 6. If (as we shall for the most part suppose) a > b > c, one of these roots (X, say) will lie between oo and - c 2 , another (fi) between — c 2 and — 6 2 , and the third (v) between — b 2 and — a 2 . The surfaces X, //,, v are therefore ellipsoids, hyperboloids of one sheet, and hyperboloids of two sheets, respectively. It follows immediately from this definition of X, fi, v, that + ft z 2 (\-O)Qi-6)(i>-0) c 2 + 6 (a 2 + 0)(b 2 + d)(c 2 + 0) •(2) a 2 +d b 2 + identically, for all values of 6. Hence multiplying by a 2 + 6, and afterwards putting 6 = — a 2 , we obtain the first of the following equations : (a 2 +X)(a 2 + ^)(a 2 + ^ (a 2 -6 2 )(a 2 -c 2 ) (b 2 + \)(b 2 +ri(b 2 + v) (b 2 -<?)(b 2 -a 2 ) 2/* = These give dx _ x x 3X~~ 2 (?Tx' 3X (c 2 + \)(c 2 + fjL)(c 2 + v) (c 2 -a 2 )(c 2 -b 2 ) ty i y dz i 6 2 + X' 3X and thence, in the notation of Art. ill (2), -* c 2 + X' .(4) 1 , ( a* y 2 h x 2 ±}( a 2 + X) 2 + (6 2 + X) 2 + .(5) (c 2 + X) 2 If we differentiate (2) with respect to 6 and afterwards put = X, we deduce the first of the following three relations : (a 2 + X)(6 2 + X)(c 2 + X) * hx 2 — 4 (X — /jl) (X — v) Aa 2_ 1 , (^ + ^(^ + ^)(c 2 + ^) {a 2 + v)(b 2 + v){c 2 + v) A 3 2 = 4 (i/ - X) (v - /a) .(6) 150 Irrotational Motion of a Liquid [chap, v The remaining relations of the sets (3) and (6) have been written down from symmetry*. Substituting in Art. Ill (4), we find -J- v '» — ( M - y )(,-x)(x-^) [»-'){^+^( y + ^^ + ^^)' + (» - X) |(a 2 + /*)* (6 2 + p)i (c 2 + M )* |- + (X - /i) |(a 2 + *)* (ft 2 + k)* (c 2 + 1,)* I!" ..:... (7) 113. The particular solutions of the transformed equation V 2 c/> = which first present themselves are those in which (f> is a function of one (only) of the variables X, fx, v. Thus <f> may be a function of X alone, provided (a 2 + X)* (b 2 + X)* (c 2 + \)i g£ = const., whence < / > = C f -r- , (1) if A = {(a 2 + X)(6 2 + \)(c 2 + X)}*, (2) the additive constant which attaches to <j> being chosen so as to make (f> vanish for X — oo . In this solution, which corresponds to <\> = A/r in spherical harmonics, the equipotential surfaces are the confocal ellipsoids, and the motion in the space external to any one of these (say that for which \ = 0) is that due to a certain arrangement of simple sources over it. The velocity at any point is given by the formula -* 2 f=4 ••••• < 3 > At a great distance from the origin the ellipsoids \ become spheres of radius \^, and the velocity is therefore ultimately equal to 20/r 2 , where r denotes the distance from the origin. Over any particular equipotential surface X, the velocity varies as the perpendicular from the centre on the tangent plane. To find the distribution of sources over the surface X = which would produce the actual motion in the external space, we substitute for <f> the value (1), in the formula (11) of Art. 58, and for <j>' (which refers to the internal space) the constant value *'=<! w * It will be noticed that h lt h 2 , h 3 are double the perpendiculars from the origin on the tangent planes to the three quadrics X, ji, v. + Cf. Lam6, "Sur les surfaces isothermes dans les corps solides homogenes en ^quilibre de emperature," Liouville, ii. 147 (1837). H2-H3] Ellipsoidal Co-ordinates 151 The formula referred to then gives, for the surface-density of the required distribution, JL h > • -< 5) The solution (1) may also be interpreted as representing the motion due to a change in the dimensions of the ellipsoid, such that the surface remains similar to itself, and retains the directions of its principal axes unchanged. If we put a/a = 6/6 = c/c, = k, say, the surface-condition Art. 110 (4) becomes — d<j>/dn = \kh 1 , which is identical with (3), if we put \ — 0, G = ^kabc. A particular case of (5) is where the sources are distributed over the elliptic disk for which A, = — c 2 , and therefore z 2 = 0. This is important in Electrostatics, but a more interesting application from the present paint of view is to the flow through an elliptic aperture, viz. if the plane xy be occupied by a thin rigid partition with the exception of the part included by the ellipse x XI -* + ^ = l> —0, we assume, putting c = in previous formulae, * = :M JoV + X)4J + X)4x* (6) where the upper limit is the positive root of and the negative or the positive sign is to be taken according as the point for which <j> is required lies on the positive or the negative side of the plane xy. The two values of <j> are continuous at the aperture, where \ = 0. As before, the velocity at a great distance is equal to 2A/r 2 , and the total flux through the area 2-rrr 2 is therefore 4tirA. The total range of <j> from X— — oo to X = + oo is 2A r * -44 \ ¥ d0 J o (a 2 + X)* (6 2 + X)* X* J o \/(a 2 sin 2 6 + 6 2 cos 2 6) The 'conductivity/ therefore, of the aperture (to borrow a term from elec- tricity) is ^•Jo V(a 2 sin 2 + 6 2 cos 2 0) W For a circular aperture this = 2a. For points in the aperture the velocity may be found immediately from (6) and (7); thus we may put x ^lA x 2 y 2 \* ^ _2A\* 152 Irrotational Motion of a Liquid [chap, v approximately, since X is small, whence -S-S'-('-S-$f <»> This becomes infinite, as we should expect, at the edge. The particular case of a circular aperture has already been solved otherwise in Arts. 102, 108. 114. We proceed to investigate the solution of V 2 ^> = 0, finite at infinity, which corresponds, for the space external to the ellipsoid, to the solution cf) = x for the internal space. Following the analogy of spherical harmonics we may assume for trial * = «X» (!) which gives V 2 ^ + -^ = 0, (2) X OX and inquire whether this can be satisfied by making % equal to some function of X only. On this supposition we shall have, by Art. Ill, d X_ h d X h fa te~ Ux' ni dx' and therefore, by Art. 112 (4), (6), 2d X = i (b* + X)(c* + X) d X ^ xdx (X — fi) (X —v) dX' On substituting the value of V 2 ^ in terms of X, the equation (2) becomes S 2 X = -(^)(c* + X)g f(a» + A.)* (6* + X)* (e» + X)* iV x = - (6 2 + X) (c* + X) & , which may be written - log {(a 2 + X)* (6 2 + X)* (c 2 + X)* ^d = - dX & ( ' . . : ^Xj a 2 + X* Hence % = cf°° s ^— i i> (3) * Ja (a 2 + X)*(6 2 +X)*(c 2 + X)* w the arbitrary constant which presents itself in the second integration being chosen as before so as to make x vanish at infinity. The solution contained in (1) and (3) enables us to find the motion of a liquid, at rest at infinity, produced by the translation of a solid ellipsoid through it, parallel to a principal axis. The notation being as before, and the ellipsoid a 2+ b 2 + c 2 W being supposed in motion parallel to x with velocity U, the surface- condition is K~*S- fo ^=° < 5 > H3-H4] Translation of an Ellipsoid 153 Let us write, for shortness, r°° d\ r°° d\ r°° d\ a ° = a H(^Tx)A' A = a6c J (^Tx)A' ^ = a6c J (?TxyA' (6) where A = {(a 2 + X) (6 2 + X) (c 2 + X)}* (7) It will be noticed that these quantities a , /3 , 7o are purely numerical. The conditions of our problem are satisfied by *-*JT(??W (8) provided Q = jbc_TJ. (9) 2 - a The corresponding solution when the ellipsoid moves parallel to y or z can be written down from symmetry, and by superposition we derive the case where the ellipsoid has any motion of translation whatever*. At a great distance from the origin, the formula (8) becomes equivalent to +-*c£. (10) which is the velocity-potential of a double source at the origin, of strength |7r(7, or compare Art. 92. The kinetic energy of the fluid is given by where I is the cosine of the angle which the normal to the surface makes with the axis of x. Since the latter integral is equal to the volume of the ellipsoid, we have ZT-^.frabcp.U* (11) The inertia-coefficient is therefore equal to the fraction *-s2j; (12) of the mass displaced by the solid. For the case of the sphere (a = b = c) we find ao = f > & = J> in agreement with Art. 92. If we put a = 6, we get the case of an ellipsoid of revolution. * This problem was first solved by Green, "Eesearches on the Vibration of Pendulums in Fluid Media," Trans. R. S. Edin. xiii. 54 (1883) [Papers, p. 315]. The investigation is much shortened if we assume at once from the Theory of Attractions that (8) is a solution of V 2 = O, being in fact (except for a constant factor) the ^-component of the attraction of a homogeneous ellipsoid at an external point. 154 Irrotational Motion of a Liquid [chap, v For the prolate ellipsoid (6=c, a> b) we find 2(l-e 2 )/,. l+e \ .... 1 i_ e 2 1 + e *>-*-?- ^r lo gfT- e ' ( 14 ) where e is the eccentricity of the meridian section. The formulae for an oblate ellipsoid are given in Art. 373. The values of k for a prolate ellipsoid moving respectively ' end-on 5 and ' broadside on,' viz. *'=ipv *-A' (15) are tabulated on the opposite page for a series of values of the ratio a/6. For an elliptic disk (a-*-0) the formula (11) becomes nugatory, since a -^2. A separate calculation, starting from (1) and (3), leads to the result 2T=§7rpb 2 c*U*+ P' J(b* sin* 6 + c* cos? 0)d6 (16) For b = c this reproduces the result (20) of Art. 102. 115. We next inquire whether the equation V 2 </> = can be satisfied by 4> = y*x> C 1 ) where % is a function of X only. This requires ^!+!!h «> Now, from Art. 112 (4), (6), 2 3% + 23x = 2/t2 /19£ + ia ? \dx ydy z dz x \y dX z dXJ dX ydy z dz \y (a 2 + \)(6 2 + X)(c 2 + X) / 1 1 \d X (\-fi)(\-v) \& 2 +\ c 2 + XjdX On substitution in (2) we find, by Art. 112 (7), whence X ~ C )k (& a + X)(c 2 +\)A' (3) the second constant of integration being chosen as before. For a rigid ellipsoid rotating about the axis of x with angular velocity £l x> the surface-condition is £-*('!->5). ^ for X == 0. Assuming * *-cy>i (i > + x)^ + x)A (5) * The expression (5) differs only by a factor from fa 9<I> where $ is the gravitation potential of a uniform solid ellipsoid at an external point (x, y, z). Since V 2 $ = it easily follows that the above is also a solution of the equation V 2 = O. 114-115] Rotation of an Ellipsoid 155 we find that the surface-condition (4) is satisfied, provided G or <7 = * ° fe + cV abc\b* -°c 2 ) " *°* (p ~ ?) (6 2 -c 2 ) 2 a^cOa; (6) 2(6 2 -c 2 ) + (6 2 + c 2 )(/3o- 7 o) The formulae for the cases of rotation about 3/ or z can be written down from symmetry*. The formula for the kinetic energy is if (Z, m, n) denote the direction-cosines of the normal to the ellipsoid. The latter integral =///(2/ 2 - *) dxdydz = J (6 2 - c 2 ) . J7ra6c. Hence we find 2T-1 (6 2 -C 2 ) 2 (70-^0) 4— .fc-'n 2 /«7X For a prolate ellipsoid (6 = c, a>6) rotating about an equatorial diameter, the ratio of the inertia coefficient to the moment of inertia, about the same diameter, of the mass of fluid displaced is found to be e 4 (/3 -«o) (2-e 2 ){2 e *-(2-* 2 )(A)-«o)} The values of k u k 2 (defined in Art. 114), and Id are shewn in the accompanying table (8) a\b *i fco k' 1 0-5 0-5 1-50 0-305 0-621 0-094 2-00 0-209 0-702 0-240 2-51 0-156 0-763 0-367 2-99 0-122 0-803 0-465 3-99 0-082 0-860 0-608 4-99 0-059 0-895 0-701 6-01 0-045 0-918 0-764 6-97 0-036 0-933 0-805 8-01 0-029 0-945 0-840 9-02 0-024 0-954 0-865 9-97 0-021 0-960 0-883 00 1 1 The two remaining types of ellipsoidal harmonic of the second order, finite at the origin, are given by the expression *2 „2 ,2 a 2 + tf + + c 2 + 6 1, •(9) * The solution contained in (5) and (6) is due to Clebsch, "Ueber die Bewegung einee Ellipsoides in einer tropfbaren Flussigkeit, " Crelle, Hi. 103, liii. 287 (1854-6). 156 Irrotational Motion of a Liquid [chap, v where 6 is either root of d^3 + W+6 + c T +6 = ' ^ 10) this being the condition that (9) should satisfy V 2 <£ = 0. The method of obtaining the corresponding solutions for the external space is explained in the treatise of Ferrers. These solutions would enable us to express the motion produced in a surrounding liquid by variations in the lengths of the axes of an ellipsoid, subject to the condition of no variation of volume : d/a + b/b + c/e=0 (11) We have already found in Art. 113, the solution for the case where the ellipsoid expands (or contracts) remaining similar to itself; so that by superposition we could obtain the case of an internal boundary changing its position and dimensions in any manner what- ever, subject only to the condition of remaining ellipsoidal. This extension of the results arrived at by Green and Clebsch was first treated, though in a different manner from that here indicated, by Bjerknes*. 116. The investigations of this chapter have related almost entirely to the case of spherical or ellipsoidal boundaries. It will be understood that solutions of the equation V 2 ^> = can be carried out, on lines more or less similar, which are appropriate to other forms of boundary. The surface which comes next in interest, from the point of view of the present subject, is that of the anchor-ring or 'torus'; this case has been very ably treated, by distinct methods, by Hicks, and Dyson f. We may also refer to the analyti- cally remarkable problem of the spherical bowl, which has been investigated by Basset J. APPENDIX TO CHAPTER V THE HYDRODYNAMICAL EQUATIONS REFERRED TO GENERAL ORTHOGONAL CO-ORDINATES "We follow the notation of Art. Ill, with this modification that differentiations of #, y, z with respect to the independent variables X, /*, v are indicated by the suffixes 1, 2, 3, respectively. Thus the direction-cosines of the normal to the surface X = const, are {h x x u h x y u Ai^i), and so on. If u, v, w be the component velocities along the three normals, the total flux out of the quasi-rectangular region whose edges are SX/Ax, 8/x/A 2 , &vjh 3 will be »(« a+ '(^) V+ »(^!) fc , ex \ h 2 n 3 J dfj.\ h z /i x J dv\ h x ti 2 J whence the expression for the expansion, viz. A-AtM, g (JL) + 1 (^) +| (^)} ; (i) cf. Art. Ill (4). * I.e. ante p. 147. f Hicks. "On Toroidal Functions," Phil. Trans, clxxii. 609 (1881); Dyson, "On the Potential of an Anchor-Ring," Phil. Trans, clxxxiv. 43 (1892); see also C. Neumann, I.e. ante p. 134. J "On the Potential of an Electrified Spherical Bowl, &c," Proc. Lond. Math. Soc. (1) xv* 286 (1885) ; Hydrodynamics, i. 149. ii5-ii6] Orthogonal Co-ordinates 157 The circulation round a rectangular circuit on the surface X = const., whose sides are §/i/A 2 , 8v/h 3 , is i(th-im»* « Dividing by the area of the circuit we get the first of the following formulae for the com- ponents of vorticity about the three normals : «-**{£©-£©}■) ,A, {sL©"c®r To find expressions for the component accelerations, we note that in a time 8t a particle changes its parameters from (X, n, v) to (X + SX, /x + fy*, v + bv\ where 8k/ h = u tit, 8[xlh 2 = vdt, 8v/h 3 =w8t. The component velocities therefore become U + \ ^+^l w 5T + ^2 v 5 _ + ^3^ ; T-)^ &C-, & c -j •(4) and we have to resolve these along the original directions of u, v, w. Now after a time 8t the direction-cosines of the new direction of v become h 2 x 2 + ^ (^2^2) hiudt + ^- (h 2 x 2 )k 2 v8t + —_ (h 2 x 2 ) h 3 w8t, &c, &c, 75 7) ?) ^- (h 2 x 2 ) hiubt + 7r- (h 2 x 2 )k 2 v8t + — Ck " OfX cv where in the two expressions not written out the derivatives of x are to be replaced by those of y and z, respectively. Hence the cosine of the angle between the new direction of v and the original direction of u, viz. (A^, Ai#i, Ai^i), is {{x x x 12 +y x y l2 + z x z 12 ) h x u + (x 1 x 22 + y x y 22 + z x z 22 ) h 2 v + fa x 23 +y t y 2 3 + *i ^23) h w) h x h 2 8t. (5) Certain terms have been omitted from this expression in virtue of the relation •^1^2+^1^2 + ^1^2 = 0, (6) which follows from the orthogonal property. Again, differentiating (6) with respect to v and comparing with similar results we infer that #1 #23+3^1 #23 + *1*23 = (7)* Also, differentiation of the identity ^2 +yi 2 + , i2== _L (8) with respect to p gives #i#i2 + ? /i#]2+zi 2 i2=T- ~ (t) W hi Cfi \tiiJ Again, ^1^22+yiy22 + 2l222 = g-(^1^2+yi3/2 + 2l22)-(^2^12 + 2/iyi2 + ^l2l2)= ~J ^ (jY .-.(10) The expression (5) thus reduces to {4,d)-4Xi)} wt (n) * Forsyth, Differential Geometry, Cambridge (1912), p. 412. 158 Orthogonal Co-ordinates [chap, v In the same way the cosine of the angle between the new direction of w and the original direction of u is {4(0-4®} m* .(12) The acceleration in the original direction of u is thus found to be du t , du , , du t , du or, more symmetrically, du dt l " l ""W , ' riV d l i l ' v ' sw di +A ^4(£)-4(£)} .(13)* 37 + A 1 w^ r +A 2 v — +A 3 w .*, {^iO'+^ts+M-i®} m The expressions for the acceleration in the direction of v and w follow by symmetry. For example, in cylindrical co-ordinates we have #=rcos#, # y = rsin0, z—z. Putting X = r, p = 0, v=z, we have A x = 1, A 2 = l/ r > A 3 =l. The expansion is accordingly du u dv dw ^Tr+r+We + lz' (15) and the components of vorticity are . dw dv s~r~dd~dz~> v- du dw dz dr , C= dv The component accelerations are du Tt +U du dr du v 2 du \ - +w y , r oz 8» dv dr dv rdd uv , dv —+w^, r dz dw Tt +U dw dr dw dw rdd oz J V ou r rd9 .(16) .(17) If in this formula we put w=0 we get the results for plane polar co-ordinates (Art. 16 a). In spherical polar s x=r sin 0coso>, y = r sin 6 sin o>, 2=rcos#. Putting X = r, fi = 0, v = a>, we have h x = 1, h 2 = 1/r, h 3 = 1/r sin 6. * G. B. Jeffery, Phil. Mag. (6) xxix. 445 (1915). APP. Orthogonal Co-ordinates 159 Hence A du u dv v 1 dw A = 3- + 2- +-57. + -cot#H r—. >. ^-, 3* r roB r r sin 6 do .(18) *-=£- 3-y rd& rsmOda* r w — cot 0, ou dw r sin 6d<o dr r .(19) The component accelerations are 3m 3m +m ^- + v 3m v 2 + w 2 3* 3r rd& rsinBdo dv dv dv dv uv w* a ot or rdO rsmBdo r r •(20) dw dw dw dw W^ U Yr +V rle^ W r sin ddo^^ WW vw , . + — + — cot 6 ; r cf. Art. 16 a. CHAPTER VI ON THE MOTION OF SOLIDS THROUGH A LIQUID: DYNAMICAL THEORY 117. In this chapter it is proposed to study the very interesting dynamical problem furnished by the motion of one or more solids in a frictionless liquid. The development of this subject is due mainly to Thomson and Tait* and to Kirchhofff. The cardinal feature of the methods followed by these writers consists in this, that the solids and the fluid are treated as forming together one dynamical system, and thus the troublesome calculation of the effect of the fluid pressures on the surfaces of the solids is avoided. To begin with the case of a single solid moving through an infinite mass of liquid, we will suppose in the first instance that the motion of the fluid is entirely due to that of the solid, and is therefore irrotational and acyclic. Some special cases of this problem have been treated incidentally in the foregoing pages, and it appeared that the whole effect of the fluid might be represented by an addition to the inertia of the solid. The same result will be found to hold in general, provided we use the term 'inertia' in a somewhat extended sense. Under the circumstances supposed, tne motion of the fluid is characterized by the existence of a single- valued velocity-potential (j> which, besides satis- fying the equation of continuity V 2 <£ = 0, (1) fulfils the following conditions : (1°) the value of —d<f>/dn, where $n denotes as usual an element of the normal at any point of the surface of the solid, drawn on the side of the fluid, must be equal to the velocity of the surface at that point normal to itself, and (2°) the differential coefficients dcf>/da), d4>/dy, d(f>/dz must vanish at an infinite distance, in every direction, from the solid. The latter condition is rendered necessary by the consideration that a finite velocity at infinity would imply an infinite kinetic energy, which could not be generated by finite forces acting for a finite time on the solid. It is also the condition to which we are led by supposing the fluid to be enclosed within a fixed vessel infinitely large and infinitely distant, all round, from the moving body. For on this supposition the space occupied by the fluid may be conceived as made up of tubes of flow which begin and end on * Natural Philosophy, Art. 320. Subsequent investigations by Lord Kelvin will be referred to later. t "Ueber die Bewegung eines Kotationskorpers in einer Fliissigkeit," Crelle, lxxi. 237 (1869) [Ges. Abh. p. 376]; Mechanik, c. xix. H7-H9] Impulse of the Motion 161 the surface of the solid, so that the total flux across any area, finite or infinite, drawn in the fluid must be finite, and therefore the velocity at infinity zero. It has been shewn in Art. 41 that under the above conditions the motion of the fluid is determinate. 118. In the further study of the problem it is convenient to follow the method introduced by Euler in the dynamics of rigid bodies, and to adopt a system of rectangular axes Ox, Oy, Oz fixed in the body, and moving with it. If the motion of the body at any instant be defined by the angular velocities p, q, r about, and the translational velocities u, v, w of the origin parallel to, the instantaneous positions of these axes *, we may write, after Kirchhoff, ^ = ^l + V^2 + ^3+^%l + g , %2 + ^%3, (2) where, as will appear immediately, <j> ly <£ 2 , </>3> %i> %2> %3 are certain functions of x, y, z determined solely by the configuration of the surface of the solid, relative to the co-ordinate axes. In fact, if I, m, n denote the direction-cosines of the normal, drawn towards the fluid, at any point of this surface, the kinematical surface-condition is rich — ^— = l(u + qz — ry) + m (v + rx — pz) + n(w+py — qx\ whence, substituting the value (2) of <£, we find -^ =n ) (3) 9 *1 _/ ^2 _ 903 r — I, ~ — fit, ~ — n, on on on 9%1 C '%2 7 3^3 7 — £- = ny — mz, — -^- = iz — nx, — -^- = mx — I 1 dn * dn dn Since these functions must also satisfy (1), and have their derivatives zero at infinity, they are completely determinate, by Art. 41 f. 119. Now whatever the motion of the solid and fluid at any instant, it might have been generated instantaneously from rest by a properly adjusted impulsive ' wrench ' applied to the solid. This wrench is in fact that which would be required to counteract the impulsive pressures pcf> on the surface, and, in addition, to generate the actual momentum of the solid. It is called by Lord Kelvin the ' impulse ' of the system at the moment under con- sideration. It is to be noted that the impulse, as thus defined, cannot be asserted to be equivalent to the total momentum of the system, which is indeed in the present problem indeterminate J. We proceed to shew however that the impulse varies, in consequence of extraneous forces acting on the solid, in exactly the same way as the momentum of a finite dynamical system. * The symbols u, v, w, p, q, r are not at present required in their former meanings, t For the particular ease of an ellipsoidal surface, their values may be written down from the results of Arts. 114, 115. J That is, the attempt to calculate it leads to 'improper' or 'indeterminate' integrals. 162 Motion of Solids through a Liquid [chap, vi Let us in the first instance consider any actual motion of a solid, from time t to time t ly under any given forces applied to it, in a finite mass of liquid enclosed by a fixed envelope of any form. Let us imagine the motion to have been generated from rest, previously to the time t , by forces (whether continuous or impulsive) applied to the solid, and to be arrested, in like manner, by forces applied to the solid after the time fa. Since the momentum of the system is null both at the beginning and at the end of this process, the time-integrals of the forces applied to the solid, together with the time-integral of the pressures exerted on the fluid by the envelope, must form an equilibrating system. The effect of these latter pressures may be calculated, by Art. 20, from the formula J-g-w+'w a> A pressure uniform over the envelope has no resultant effect ; hence, since <j> is constant at the beginning and end, the only effective part of the integral pressure fp dt is given by the term -iplq'dt (2) Let us now revert to the original form of our problem, and suppose the containing envelope to be infinitely large, and infinitely distant in every direction from the moving solid. It is easily seen by considering the arrange- ment of the tubes of flow (Art. 36) that the fluid velocity q at a great distance r from an origin in the neighbourhood of the solid will ultimately be, at most*, of the order 1/r 2 , and the integral pressure (2) therefore of the order 1/r 4 . Since the surface-elements of the envelope are of the order r^-sr, where Scr is an elementary solid angle, the force- and couple-resultants of the integral pressure (2) will now both be null. The same statement therefore holds with regard to the time-integral of the forces applied to the solid. If we imagine the motion to have been started instantaneously at time t , and to be arrested instantaneously at time t 1} the result at which we have arrived may be stated as follows : The ' impulse ' of the motion (in Lord Kelvin's sense) at time t± differs from the ' impulse ' at time t by the time-integral of the extraneous forces acting on the solid during the interval t\ — t f. It will be noticed that the above reasoning is substantially unaltered when the single solid is replaced by a group of solids, which may moreover be flexible instead of rigid, and even when these solids are replaced by masses of liquid which are moving rotationally. 120. To express the above result analytically, let f, rj, f, \, /jl, v be the components of the force- and couple-constituents of the impulse ; and let * It is really of the order 1/r 3 when, as in the case considered, the total flux outwards is zero, t Sir W. Thomson, I.e. ante p. 33. The form of the argument given above was kindly- suggested to the author by Sir J. Larmor. ii9-i2i] Kinetic Energy 163 X, Y, Z, L, M, N designate in the same manner the system of extraneous forces. The whole variation of £, rj, J, X, yu., v, due partly to the motion of the axes to which these quantities are referred, and partly to the action of the extraneous forces, is then given by the formulae* -i=r/7-g?+X, ^- = wrj - v£ + rp- qv + Z, ^ g = p f - r £ + F, d £=uZ-wZ+pp-r\ + M,\ (1) TO 7 -± = q£- pv + Z, ~ = v£-u V J-q\-pfx + N. For at time t + 8t the moving axes make with their positions at time t angles whose cosines are (1, r8t, -qSt), (-r8t, 1, p8t), (q8t, -p8t, 1), respectively. Hence, resolving parallel to the new position of the axis of x y £ + 8f;=!; + v.rSt-Z.q$t + X8t. Again, taking moments about the new position of Ox, and remembering that has been displaced through spaces u8t, v8t, w8t parallel to the axes, we find \ + 8\ = \ + r).w8t — %.vht + /j..r8t — v .q8t + L8t. These, with the similar results which can be written down from symmetry, give the equations (1). When no extraneous forces act, we verify at once that these equations have the integrals f a + V 2 + ? 2 = const., \% + m + v£= const., (2) which express that the magnitudes of the force- and couple-resultants of the impulse are constant. 121. It remains to express f, rj, f, \, /jl, v in terms of u, v, w, p, q, r. In the first place let T denote the kinetic energy of the fluid, so that 2T ->li*£* « where the integration extends over the surface of the moving solid. Substi- tuting the value of </> from Art. 118 (2), we get 2T = Aw 2 + Bv 2 + Cw 2 + 2A'vw + 2B' wu + 2C'uv + Tp 2 + Qq 2 + Rr 2 + 2T'qr + 2Q' rp + 2R'pq + 2p (Fu + Gv + Hw) + 2q (F'u + G'v + U'w) + 2r (F"u + G"v + H"w), (2) where the twenty-one coefficients A, B, C, &c. are certain constants * Cf. Hayward, " On a Direct Method of Estimating Velocities, Accelerations, and all similar Quantities, with respect to Axes moveable in any manner in space," Camb. Trans, x. 1 (1856). 164 Motion of Solids through a Liquid [chap, vi determined by the form and position of the surface relative to the co-ordinate axes. Thus, for example, A A \ -■-*JJ*£*~'JJ*&" •(3) = p \\ fan dS — p <j> z m dS, * — P //» ^ dS = P jjxi (Jiy - mz) dS, i the transformations depending on Art. 118 (3) and on a particular case of Green's Theorem (Art. 44 (2)). These expressions for the coefficients were given by Kirch hofT. The actual values of the coefficients in the expression for 2T have been found in the preceding chapter for the case of the ellipsoid, viz. we have from Arts. 114, 115 _«c 2-a A = ^- .^npabc, (& 2 -c 2 ) 2 (yo-/3 ) 5 2(&2- C 2) + (62 + c 2 )(/3() _ 7o ) . I npabc, •(4) with similar expressions for B, 0, Q, R. The remaining coefficients, as will appear pre- sently, in this case all vanish. We note that 2(a -/3o) A-B= . ^ npabc, .(5) (2-« )(2-/3 ) so that if a > b > c, then A< B < O, as might have been anticipated. The formulae for an ellipsoid of revolution may be deduced by putting b = c ; they may also be obtained independently by the method of Arts. 104-109. Thus for a circular disk (a=0, b = c) we have A, B, C = f pC 3, 0, 0; P, Q, R = 0, l%pc\ ^P<* (6) 121 a. When the motion of the solid is one of pure translation the formula for the kinetic energy of the fluid reduces to 2T = Au 2 + Bv 2 +Cw 2 + 2A'vw + 2B'wu + 2C'uv (1) We can now shew that the effect at a great distance is in all cases that of a suitable double source, and that the character of this source is completely defined by the coefficients in (1). For this we have recourse to the formula (12) of Art. 58, viz. ,J 1 Wp-Jjtf-^-dS. ■(2) We may regard the boundary of the solid as a thin rigid shell, with fluid also in its interior, and assume the potentials </> and cp' to refer to the external 121-121 a] Effect at a Distance 165 and internal regions, respectively. Let (x 1} y 1} z-±) be the co-ordinates of the point P, which we suppose to be at a distance great compared with the dimensions of the solid, and (x, y, z) those of a surface-element SS. Then, writing n = V(% 2 + 2/! 2 + z?\ r = V{(^i - xf + (y x - yf + {z x - zf}, we have, approximately, 1 _ 1 xx-i + yy x + zz-i 3 1_ lx\ + my x 4- nz x r Tx ^i 3 ' dn r rj? Suppose, now, that the shell is moving with unit velocity parallel to x, without rotation. Writing = 1? f = -*, (3) u {{j. dl jv Ax 1 + C'y 1 + B'z 1 ... we have \\6 = dS = — ^ , (4) JJ^dnr prf and JJ* r s>~£- < 5 > where Q denotes the volume of the solid. We have, in fact, llwldS=Q, ljxmdS = 0, llxndS = (6) Hence ^ ^ p JA±PQ^l±pl±^3 (7) * The effect at a distance is therefore that of a double source, but the axis of the source does not necessarily coincide with the direction of transition. If, however, the solid is moving parallel to an axis of permanent translation (Art. 124), the coefficients C and B' vanish, and 4^ = <A±^ (8) For example, in the case of the sphere we have A = §7r/oa 3 , Q = |7ra 3 , and ^ = 2^' (9) as in Art. 92. When the velocity (u, v, w) of the solid is general, the formula (7) is replaced by 47rr 1 3 / o0p = (Au + C'v + B'w) x x + (G'u + Bv + A'w) y x \ {B'u + A'v + Cw) z x + pQ(ux 1 + vy 1 + wz-s) (10) Conversely a knowledge of the form of the velocity-potential at infinity due to a ' per- manent ' translation leads to a knowledge of the corresponding inertia-coefficient. * From a paper "On Wave Eesistance," Proc. Roy. Soc. cxi. 15 (1926). 166 Motion of Solids through a Liquid [chap, vi For instance, in the Rankine ovals referred to in Art. 97 we have a distribution of sources along the axis of x, subject to the condition that the total 'strength' of these sources is zero. If the line-density of this distribution be m, we have V J >/{{*! - 1) 2 +yi 2 + *i 2 } J Vi rf J g ' or #-^ J **£<% + ..-, (11) since jmd£=0. Hence A/p + Q=47rSm£d£ (12)* 122. The kinetic energy, Ti say, of the solid alone is given by an expres- sion of the form + Pii> 2 + Qiq 2 + Rir 2 -r 2PiV + 2Qi'rp + 2R 1 'pq + 2m {a (vr — wq) + /? (wp — ur) + y (uq — vp)} (1) Hence the total energy T+-Ti, of the system, which we shall denote by T, is given by an expression of the same general form as in Art. 121, say 2T = An 2 + Bv 2 + Gw 2 + 2A'vw + 2B'wu + 2C'uv + Pp 2 + Qq 2 + Rr 2 + 2Fqr + 2Q'rp + 2R'pq + 2p (Fu + Gv + Hw) + 2q (F'u + G'v + H'w) + 2r {F"u + G"v + H "w). (2) The values of the several components of the impulse in terms of the velo- cities u, v, w, p, q, r can now be found by a well-known dynamical method f. Let a system of indefinitely great forces (X, Y, Z, L, M, N) act for an indefinitely short time t on the solid, so as to change the impulse from (ft 17, f, \, fi, v) to (? + S£ v + H ?+8£ \ + B\, /i + S/x, * + &/). The work done by the force X, viz. \ T Xudt, J o lies between % X cfa and u 2 X dt, Jo Jo where ui and u 2 are the greatest and least values of u during the time t, i.e. it lies between %8f and u 2 B^. If we now introduce the supposition that Sf, S77, Sf, 8\, $/z, 81/ are infinitely small, % and u 2 are each equal to u, and the work done is uB%. In the same way we may calculate the work done by the remaining forces and couples. The total result must be equal to the increment of the kinetic energy, whence u 8f + v Brj + w 8f + p BX + q Bfi + r Bv = BT^Su + d ^8v + ^Sw + d ^Bp + fs q + d ^Sr. ...(3) ou ov ow dp * oq or * G. I. Taylor, Proc. Roy. Soc, cxx. 13 (1928). t See Thomson and Tait, Art. 313, or Maxwell, Electricity and Magnetism, Part iv. c. v. i2ia-i22] Relations between Energy and Impulse 167 Now if the velocities be all altered in any given ratio, the impulses will be altered in the same ratio. If then we take 8u 8v u V 8w 8p w p _ B 2 _ Br _ k q r it will follow that S£=h = 8£ 8\ hfj, 8v , /J, V Substituting in (3), we find u!; + vt) + w%+p\ + qp + rv dT dT dT dT , dT dT om ... = u ^ + v ^ + w ^ + P^ + lY q +r ^ =2f ' -< 4 > since T is a homogeneous quadratic function. Now performing the arbitrary variation 8 on the first and last members of (4), and omitting terms which cancel by (3), we find f$w + v 8v + %8w + \8p + fi8q + v8r = 8T. Since the variations 8u, 8v, 8iu, 8p, 8q, 8r are all independent, this gives the required formulae „ dT dT dT % dT dT dT /KX ?' * ?= ^' *i> di>> X ' * " = ^' dq> SF (5) It may be noted that since jf , 17, J, . . . are linear functions of w, v, w, . . . , the latter quantities may also be expressed as linear functions of the former, so that T may be regarded as a homogeneous quadratic function of f, 77, f, \, /jl, v. When expressed in this manner we may denote it by T\ The equation (3) then gives at once u 8f + v 8rj + w 8£ + p 8\ + q 8/j, + r 8v dT\ t dT\ ^\ y .dT"dT\ dT\ dT" dr dT s dr dT dr whence „,*, w = __,_, ^, g , r = __,__, __ ( 6 ) These formulae are in a sense reciprocal to (5). We can utilize this last result to obtain, when no extraneous forces act, another integral of the equations of motion, in addition to those found in Art. 120. Thus dt ~ d% dt + '" + '" + ax dt^'"^'" d% dX which vanishes identically, by Art. 120 (1). Hence we have the equation of energy T=const ...(7) 168 Motion of Solids through a Liquid [chap, vi 123. If in the formulae (5) we put, in the notation of Art. 121, T=T + T 1 , it is known from the Dynamics of rigid bodies that the terms in T x represent the linear and angular momentum of the solid by itself. Hence the remaining terms, involving T, must represent the system of impulsive pressures exerted by the surface of the solid on the fluid, in the supposed instantaneous genera- tion of the motion from rest. This is easily verified. For example, the ^-component of the above system of impulsive pressures is = Au + C'v + B'w + Pp + ¥'q + F'V 8T .(8) by the formulae of Arts. 118, 121. In the same way, the moment of the impulsive pressures about Ox is dxi ))p<t>( n y mz) dS Fu + Gv + Kw + Pp + H'q + Q V *~£' dS dr dp ' (9) 124. The equations of motion may now be written* ddT dt du ddT dt dv d dT dT dT _ dv dT dw \ ^ dw du ' dT dT dt dw ^ du P dv d dT dT dT dT dt dp dv dw dq + Z, dT T ^d~r +L ' (1) ddT dt dq dT dw ±dT = dT dt dr du dT du dT T + ^ = u ^- w *r+P^:- r ^ + M > dT dr dT dv ^dp P dT dp dT + N. If in these we write T= T + Ti, and isolate the terms due to T, we obtain expressions for the forces exerted on the moving solid by the pressure of the surrounding fluid; thus the total component (X, say) of the fluid pressure parallel to x is ddT dT dT X = dt du, dv dw' •(2) * See Kirchhoff, I.e. ante p. 160; also Sir W. Thomson, " Hypokinetic Solutions and Obser- vations," Phil. Blag. (5) xlii. 362 (1871) [reprinted in Baltimore Lectures, Cambridge, 1904, p. 584]. 123-124] Equations of Motion 169 and the moment (L) of the same pressures about x is * dt dp dv dw dq ^ dr' For example, if the solid be constrained to move with a constant velocity (u, v, w\ without rotation, we have X, Y, Z - 0, ) T nwr ^ ST dT dT dT dT dT \ (4) L, M, N = w ^ # — , u^ w x- , Vr u-^- , dv dw dw du du dv ) where 2T = Au 2 + Bv 2 + Cw 2 + 2A'vw + 2B' wu + 2C'uv. The fluid pressures thus reduce to a couple, which moreover vanishes if dT = aT _3T du' dv ' dw' ' i.e. provided the velocity (u, v, w) be in the direction of one of the principal axes of the ellipsoid Ax 2 + By 2 + Cz 2 + 2A'yz + ZB'zx + 2C'xy = const (5) Hence, as was first pointed out by Kirchhoff, there are, for any solid, three mutually perpendicular directions of permanent translation; that is to say, if the solid be set in motion parallel to one of these, without rotation, and left to itself, it will continue to move in this manner. It is evident that these directions are determined solely by the configuration of the surface of the body. It must be observed however that the impulse necessary to produce one of these permanent translations does not in general reduce to a single force ; thus if the axes of co-ordinates be chosen, for simplicity, parallel to the three directions in question, so that A' t B r , C = 0, we have, corresponding to the motion u alone, f, v , Z=Au, 0, 0; X, //,, v—Fu, F'u, F"u, so that the impulse consists of a wrench of pitch FjA. With the same choice of axes, the components of the couple which is the equivalent of the fluid pressures on the solid, in the case of any uniform translation (u, v, w), are L, M, N = (B-C)w;, (C-A)ww, (A-B)w; (6) Hence if in the ellipsoid Ax 2 + By 2 +Cz 2 = const., (7) we draw a radius vector r in the direction of the velocity (u, v, w) and erect the perpendicular h from the centre on the tangent plane at the extremity of r, the plane of the couple is that of h and r, its magnitude is proportional to sin (h, r)/h, and its tendency is to turn the solid in the direction from h to r. * The forms of these expressions being known, it is not difficult to verify them by direct calculation from the pressure-equation, Art. 20 (5). See a paper "On the Forces experienced by a Solid moving through a Liquid," Quart. Journ. Math. xix. 66 (1883). 170 Motion of Solids through a Liquid [chap, vi Thus if the direction of (u, v, w) differs but slightly from that of the axis of x, the tendency of the couple is to diminish the deviation when A is the greatest, and to increase it when A is the least, of the three quantities A, B, C, whilst if A is intermediate to B and C the tendency depends on the position of r relative to the circular sections of the above ellipsoid. It appears then that of the three permanent translations one only is thoroughly stable, viz. that corresponding to the greatest of the three coefficients A, B, C. For example, the only stable direction of translation of an ellipsoid is that of its least axis; see Art. 121*. 125. The above, although the simplest, are not the only steady motions of which the body is capable, under the action of no extraneous forces. The instantaneous motion of the body at any instant consists, by a well-known theorem of Kinematics, of a twist about a certain screw ; and the condition that this motion should be permanent is that it should not affect the configuration of the impulse (which is fixed in space) relatively to the body. This requires that the axes of the screw and of the corresponding impulsive wrench should coincide. Since the general equations of a straight line involve four independent constants, this gives four linear relations to be satisfied by the five ratios u : v; w: p : q : r. There exists then for every body, under the circumstances here considered, a singly-infinite system of possible steady motions. The steady motions next in importance to the three permanent translations are those in which the impulse reduces to a couple. The equations (1) of Art. 120 shew that we may have £, 77, £= 0, and X, p, v constant, provided X/p=fM/q= v /r, =k, say (1) If the axes of co-ordinates have the special directions referred to in the preceding Art., the conditions £, 77, £=0 give us at once u, v, w in terms of p, q, r, viz. Fp + F'q + F"r Gp + G'q + G"r Hp + H'q+H"r u ~ A ' V ~ B ' W C () Substituting these values in the expressions for X, /u, v obtained from Art. 122 (5), we find de de de *>» v =ty> ty> «■' < 3 > provided 29 (p, q, r) = typ 2 + i&q 2 + %ir 2 + 2Wqr + 2i$rp+2Wpq, (4) the coefficients in this expression being determined by formulae of the types m-P-^1-^-^ m P> *"*"' Q ' G " H ' H " ^ ** A B C ®~ A B C W These formulae hold for any case in which the force-constituent of the impulse is zero. Introducing the conditions (1) of steady motion, the ratios p : q : r are to be determined from the three equations \Bp + &q + <B'r = kp,^ Wp + ®q + W'r=kq\ (6) <&p + '$q+1&r = kr. J * The physical cause of this tendency of an elongated body to set itself broadside-on to the relative motion is clearly indicated in the diagram on p. 86. A number of interesting practical illustrations are given by Thomson and Tait, Art. 325. 124-125] Steady Motions 171 The form of these shews that the line whose direction-ratios are p:q :r must be parallel to one of the principal axes of the ellipsoid Q(x, y, 2) = const (7) There are therefore three permanent screw-motions such that the corresponding impulsive wrench in each case reduces to a couple only. The axes of these three screws are mutually at right angles, but do not in general intersect. It may now be shewn that in all cases where the impulse reduces to a couple only, the motion can be completely determined. It is convenient, retaining the same directions of the axes, to change the origin. Now the origin may be transferred to any point (#, y, z) by writing u + ry-qz, v + pz — rx, w + qx-py, for u, v, w respectively. The coefficient of 2vr in the expression for the kinetic energy, Art. 122 (2), becomes -Bx+G", that of 2wq becomes Cx + H\ and so on. Hence if we take X (G" E'\ ,(H F"\ (F' 0\ ... the coefficients in the transformed expression for 2T will satisfy the relations <r_#' h__f^_ F l_ ( * B~C' C~A' A~ B (y) If we denote the values of these pairs of equal quantities by a, /3, y respectively, the formulae (2) may be written d* d* dV ,„,_. M= -^' v= ~dj' w= ~w (10) where 2* ( p, q, r) =s — p 2 + -^ q 2 -f — ^ r 2 + 2aqr + 2ftrp + 2ypq (11) The motion of the body at any instant may be conceived as made up of two parts ; viz. a motion of translation equal to that of the origin, and one of rotation about an instantaneous axis passing through the origin. Since £, 77, (=0 the latter part is to be determined by the equations dX du, . dv T =r v .-qy i f t =pv-r\ g -flX-W which express that the vector (X, /x, v) is constant in magnitude and has a fixed direction in space. Substituting from (3), d L de_ 9e_ de \ dt dp~ dq ^ dr ' d de de de dtTq=VTr~ r irp^ ' {l " } d_de_ de_ de dt dr~^dp ™dq These are identical in form with the equations of motion of a rigid body about a fixed point, so that we may make use of Poinsot's well-known solution of the latter problem. The angular motion of the body is obtained by making the ellipsoid (7), which is fixed in the body, roll on a plane \x + fxy + vz = const. , which is fixed in space, with an angular velocity proportional to the length 01 of the radius vector drawn from the origin to the point of contact 7. The representation of the actual motion is then completed by impressing on the whole system of rolling ellipsoid 172 Motion of Solids through a Liquid [chap, vi and plane a velocity of translation whose components are given by (10). This velocity is in the direction of the normal M to the tangent plane of the quadric *tay,#)-T-«», (13) at the point P where 01 meets it, and is equal to <? 3 np DM xam ? u l ar velocity of body (14) When 0/does not meet the quadric (13), but the conjugate quadric obtained by changing the sign of e, the sense of the velocity (14) is reversed*. 126. The problem of the integration of the equations of motion of a solid in the general case has engaged the attention of several mathematicians, but, as might be anticipated from the complexity of the question, the physical meaning of the results is not easily grasped f. In what follows we shall in the first place inquire what simplifications occur in the formula for the kinetic energy, for special classes of solids, and then proceed to investigate one or two particular problems of considerable interest which can be treated without difficult mathematics. The general expression for the kinetic energy contains, as Ave have seen, twenty-one coefficients, but by the choice of special directions for the co-ordinate axes, and a special origin, these can be reduced to fifteen J. The most symmetrical way of writing the general expression is 2T = An 2 + Bv 2 + Cw 2 + 2A'vto + 2B'wu + 2C'uv + Pp 2 + Qq 2 + Rr 2 + 2P'qr + 2Q'rp + 2R'pq + 2Lup + 2Mvq + 2Nwr + 2F(vr + wq) + 2G (wp + ur) + 2H(uq + vp) + 2F' (vr-wq)+2G' (wp-ur) + 2H' (uq-vp) (1) It has been seen that we may choose the directions of the axes so that A\ B' ', C = 0, and it may easily be verified that by displacing the origin we can further make F\ G' } H' = 0. We shall henceforward suppose these simplifications to have been made. 1°. If the solid has a plane of symmetry, it is evident from the con- figuration of the relative stream-lines that a translation normal to this plane must be one of the permanent translations of Art. 124. If we take this plane as that of xy y it is further evident that the energy of the motion must be unaltered if we reverse the signs of w, p, q. This requires that P', Q', L, M, N, H should vanish. The three screws of Art. 125 are now pure rotations, but their axes do not in general intersect. * The substance of this Art. is taken from a paper, "On the Free Motion of a Solid through an Infinite Mass of Liquid," Proc. Lond. Math. Soc. viii. 273 (1877). Similar results were obtained independently by Craig,." The Motion of a Solid in a Fluid," Amer. Journ. of Math. ii. 162 (1879). f For references see Wien, Lehrbuch d. Hydrodynamik, Leipzig, 1900, p. 164. J Cf. Clebsch, "Ueber die Bewegung eines Korpers in einer Flussigkeit," Math. Ann. hi. 238 (1870). This paper deals with the 'reciprocal' form of the dynamical equations, obtained by substituting from Art. 122 (6) in Art. 120 (1). 125-126] HydroMnetic Symmetries 173 2°. If the body has a second plane of symmetry, at right angles to the former one, we may take this as the plane xz. We find that in this case R' and G must also vanish, so that 2T = Au 2 + Bv 2 +Cw 2 + Pp 2 + Qq 2 + Rr 2 + 2F(vr + wq) (2) The axis of x is the axis of one of the permanent rotations, and those of the other two intersect it at right angles, though not necessarily in the same point. 3°. If the body has a third plane of symmetry, say that of yz, at right angles to the two former ones, we have 2T = Au 2 + Bv 2 + Gw 2 + Pp 2 +Qq 2 + Rr 2 (3) 4°. Returning to (2°), we note that in the case of a solid of revolution about Ox, the expression for 2T must be unaltered when we write v,q, — w, — r for w, r, v, q, respectively, since this is equivalent to rotating the axes of y, z through a right angle. Hence B= G, Q = R, F=0; and therefore 2T = Au 2 + B(v 2 + w 2 ) + Pp 2 + Q(q 2 + r 2 ) (4)* The same reduction obtains in some other cases, for example when the solid is a right prism whose section is any regular polygon f. This is seen at once from the consideration that, the axis of x coinciding with the axis of the prism, it is impossible to assign any uniquely symmetrical directions to the axes of y and z. 5°. If, in the last case, the form of the solid be similarly related to each of the co-ordinate planes (for example a sphere, or a cube), the expression (3) takes the form 2T = A(u 2 +v 2 + w 2 ) + P(p 2 + q 2 + r 2 ) (5) This again may be extended, for a like reason, to other cases, for example any regular polyhedron. Such a body is practically for the present purpose 'isotropic,' and its motion will be exactly that of a sphere under similar conditions. 6°. We may next consider another class of cases. Let us suppose that the body has a sort of skew symmetry about a certain axis (say that of x), viz. that it is identical with itself turned through two right angles about this axis, but has not necessarily a plane of symmetry]:. The expression for 2T must be unaltered when we change the signs of v, w, q, r, so that the coefficients Q\ R', G, H must all vanish. We have then 2T = An 2 + Bv 2 + Gw 2 + Pp 2 + Qq 2 + Rr 2 + 2P'qr + 2Lup + 2Mvq + 2Nwr 4- 2F (vr + wq) (6) * For the solution of the equations of motion in this case see Greenhill, "The Motion of a Solid in Infinite Liquid under no Forces," Amer. Journ. of Math. xx. 1 (1897). t SeeLarmor, "On Hydrokinetic Symmetry," Quart. Journ. Math. xx. 261 (1884). [Papers, i. 77.] { A two-bladed screw-propeller of a ship is an example of a body of this kind. 174 Motion of Solids through a Liquid [chap, vi The axis of x is one of the directions of permanent translation ; and is also the axis of one of the three screws of Art. 125, the pitch being —L/A. The axes of the two remaining screws intersect it at right angles, but not in general in the same point. 7°. If, further, the body be identical with itself turned through one right angle about the above axis, the expression (6) must be unaltered when v, q, —w, —r are written for w, r, v, q, respectively. This requires that B=C,Q = R,P' = 0,M = N,F=0. Hence* 2T = A u 2 + B (v 2 + w 2 ) +Pp 2 + Q(q 2 + r 2 ) + 2Lup + 2M (vq + wr). . . .(7) The form of this expression is unaltered when the axes of y, z are turned in their own plane through any angle. The body is therefore said to possess helicoidal symmetry about the axis of x. 8°. If the body possess the same properties of skew symmetry about an axis intersecting the former one at right angles, we must evidently have 2T=A(u 2 + v 2 + w 2 ) + P(p 2 + q 2 + r 2 )+2L(pu + qv + rw). ...(8) Any direction is now one of permanent translation, and any line drawn through the origin is the axis of a screw of the kind considered in Art. 125, of pitch —L/A. The form of (8) is unaltered by any change in the directions of the axes of co-ordinates. The solid is therefore in this case said to be ' helicoidal ly isotropic' 127. For the case of a solid of revolution, or of any other form to which the formula 2T= Au 2 + B (v 2 + w 2 ) + Pp 2 + Q (q 2 + r 2 ) (1) applies, the complete integration of the equations of motion was effected by Kirchhofff in terms of elliptic functions. The particular case where the solid moves without rotation about its axis, and with this axis always in one plane, admits of very simple treatment!, and the results are very interesting. If the fixed plane in question be that of xy we have p, q, w = 0, so that the equations of motion, Art. 124 (1), reduce to A- T - = rBv, B-z- = —rAu, at dt ' , ! 12) Let x, y be the co-ordinates of the moving origin relative to fixed axes in the plane {xy) in which the axis of the solid moves, the axis of x coinciding with the line of the * This result admits of the same kind of generalization as (4), e.g. it applies to a body- shaped like a screw-propeller with three symmetrically-disposed blades. The integration of the equations of motion is discussed by Greenhill, "The Motion of a Solid in Infinite Liquid," Amer. Journ. of Math, xxviii. 71 (1906). t I.e. ante p. 160. % See Thomson and Tait, Art. 322; Greenhill, "On the Motion of a Cylinder through a Frictionless Liquid under no Forces," Mess, of Math. ix. 117 (1880). 126-127] Solid of Revolution 1 75 resultant impulse (/, say) of the motion ; and let 6 be the angle which the line Ox (fixed in the solid) makes with X. We have then Au = Icoa8, Bv= — isintf, r=6. The first two of equations (2) merely express the fixity of the direction of the impulse in space ; the third gives Q'6+ A ^I 2 sin6cos6 = (3) We may suppose, without loss of generality, that A>B. If we write 26 = $, (3) becomes ^Tf*^ (4) which is the equation of motion of the common pendulum. Hence the angular motion of the body is that of a ' quadrantal pendulum,' i.e. a body whose motion follows the same law in regard to a quadrant as the ordinary pendulum does in regard to a half-circum- ference. When 8 has been determined from (3) and the initial conditions, X, y are to be found from the equations X = u cos 6 — v sin 8 = — cos 2 6 + -75 sin 2 8, .(5) (I I\ Q .. -j — -£ ) sin 8 cos 8= yd, the latter of which gives y=jt, (6) as is otherwise obvious, the additive constant being zero since the axis of X is taken to be coincident with, and not merely parallel to, the line of the impulse /. Let us first suppose that the body makes complete revolutions, in which case the first integral of (3) is of the form 6 2 =<& 2 (l-k 2 sm 2 0), (7) where * 2= w3 (8) Hence, reckoning t from the position 0=0, we have *<<^Wi=w>' < 9 > in the usual notation of elliptic integrals. If we eliminate t between (5) and (7), and then integrate with respect to 6, we find {-L + t) f ^~t £ ^^ .(10) the origin of X being taken to correspond to the position 0=0. The path can then be traced, in any particular case, by means of Legendre's Tables. See the curve marked I on the next page. If, on the other hand, the solid does not make a complete revolution, but oscillates through an angle a on each side of the position 6 = 0, the proper form of the first integral of (3) is M x -S5' < n > , . „ ABO a> 2 where sma== zri-72 (12) 176 If we put this gives whence Motion of Solids through a Liquid [chap, vi sin 6 = sin a sin yjr, ^ 2= ^ « ( X ~ sin2 a sin2 +)> = i^(sino,Vr). sin 2 a cot sin a 127-128] Solid of Revolution 111 Transforming to yjr as independent variable, in (5), and integrating, we find .(14) .(17) X= -=- sin a . F (sin a, \ls) — -V cosec a . E (sin a, \i/-), £><£> 1 y=-V cos^. The path of the point is now a sinuous curve crossing the line of the impulse at intervals of time equal to a half-period of the angular motion. This is illustrated by the curves III and IV of the figure. There remains a critical case between the two preceding, where the solid just makes a half-revolution, 6 having as asymptotic limits the two values ±^tt. This case may be obtained by putting k= 1 in (7), or a.=%ir in (1 1) ; and we find = cocos<9, (15) o>*=log tan (J «■ + ££), (16) x= 7r -logtan (i7r + £0)--^sin0, y=^cos0. See the curve II of the figure*. It is to be observed that the above investigation is not restricted to the case of a solid of revolution ; it applies equally well to a body with two perpendicular planes of sym- metry, moving parallel to one of these planes, provided the origin be properly chosen. If the plane in question be that of xy, then on transferring the origin to the point (FjB, 0, 0) the last term in the formula (2) of Art. 126 disappears, and the equations of motion take the form (2) above. On the other hand, if the motion be parallel to zx we must transfer the origin to the point ( - F/C, 0, 0). The results of this Article, with the accompanying diagram, serve to exemplify the statements made near the end of Art. 124. Thus the curve IV illustrates, with exaggerated amplitude, the case of a slightly disturbed stable steady motion parallel to an axis of per- manent translation. The case of a slightly disturbed unstable steady motion would be represented by a curve contiguous to II, on one side or the other, according to the nature of the disturbance. 128. The mere question of the stability of the motion of a body parallel to an axis of symmetry may of course be treated more simply by approximate methods. Thus, in the case of a body with three planes of symmetry, as in Art. 126, 3°, slightly disturbed from a state of steady motion parallel to x, we find, writing u = u -f u, and assuming u', v, w, p, q, r to be all small, A du' _ „dv . „dw . \ A w= ' B dt = - Au « r ' °-dt^ Au ^ I (1) * In order to bring out the peculiar features of the motion, the curves have been drawn for the somewhat extreme case of A=5B. In the case of an infinitely thin disk, without inertia of its own, we should have A/B = co ; the curves would then have cusps where they meet the axis of y. It appears from (5) that x has always the same sign, so that loops cannot occur in any case. In the various cases figured the body is projected always with the same impulse, but with different degrees of rotation. In the curve I, the maximum angular velocity is J2 times what it is in the critical case II ; whilst the curves III and IV represent oscillations of amplitude 45° and 18° respectively. 178 Motion of Solids through a Liquid [chap, vi „ 6 e2V A(A-B) 2 . Hence ^~n&^ ~~R -tioV = 0, with a similar equation for r, and n d 2 w A(A-G) 2 . , ox a ^ + ^Q--V^ = 0, (2) with a similar equation for ^. The motion is therefore stable only when A is the greatest of the three quantities A, B, C. It is evident from ordinary Dynamics that the stability of a body moving parallel to an axis of symmetry will be increased, or its instability (as the case may be) will be diminished, by communicating to it a rotation about this axis. This question has been examined by Greenhill*. Thus, in the case of a solid of revolution slightly disturbed from a state of motion in which u and p are constant and the remaining velocities are zero, if we neglect squares and products of small quantities the first and fourth of equations (1) of Art. 124 give du/dt = 0, dp/dt**0, whence u = u , P=Po, (3) say, where w , p are constants. The remaining equations then take, on substitution from Art. 126 (3), the forms B \d~ P ° W ) = ~ A M ° r ' B \dt + p ° v ) = A U ° q ' •(4) Q^ t +(P-Q)por=-(A-B)uoW, Q ( t-(P-Q)p q = (A-B)u v (5) If we assume that v, w, q, r vary as e i<Tt , and eliminate their ratios, we find Q^±(P-2Q) Po cr-i [ (P-Q)p^ + ~(A-B)u ( ^=0 (6) The condition that the roots of this should be real is that PW + 4JjU-B)Qu i should be positive. This is always satisfied when A>B, and can be satisfied in any case by giving a sufficiently great value to p . This example illustrates the steadiness of flight which is given to an elongated projectile by rifling. 129. In the investigation of Art. 125 the term 'steady' was used to characterize modes of motion in which the ' instantaneous screw ' preserved a constant relation to the moving solid. In the case of a solid of revolution, however, we may conveniently use the term in a somewhat wider sense, extending it to motions in which the vectors representing the velocities of translation and rotation are of constant magnitude, and make constant angles with the axis of symmetry and with each other, although their relation to points of the solid not on the axis may continually vary. * "Fluid Motion between Confocal Elliptic Cylinders, &c." Quart. Journ. Math. xvi. 227 (1879). 128-130] Stability 179 The conditions to be satisfied in this case are most easily obtained from the equations of motion of Art. 124, which become, on substitution from Art. 126 (4), dp .(1) A^B(rv-qw), P^ = 0, B^=Bpw-Aru, Q^=-(A-B)uw-(P-Q)pr, B^^Aqu-Bpv, Q^ = (A-B)uv + (P-Q)pq.^ It appears that p is in any case constant, and that q 2 + r 2 will also be constant provided v/q=wr, = &, say (2) This makes du/dt—0, and v 2 + w 2 = const. It follows that k will also be constant; and it only remains to satisfy the equations kB^{kBp-Au)r, Q ( k=-{( K A-B)ku + {P-Q)p}r. These will be consistent provided kB{(A-B)ku + (P-Q)p} + Q(kBp-Au) = 0, , u kBP /0 . whence -=T7i — 7 o P/ A — m ( 3 ) p AQ-k 2 B(A-B) Hence by variation of k we obtain an infinite number of possible modes of steady motion, of the kind above defined. In each of these the instantaneous axis of rotation and the direction of translation of the origin are in one plane with the axis of the solid. It is easily seen that the origin describes a helix about the line of the impulse. These results are due to KirchhofF. 130. The only case of a body possessing helicoidal property, where simple results can be obtained, is that of the 'isotropic helicoid ' defined by Art. 126 (8). Let be the centre of the body, and let us take as axes of co-ordinates at any instant a line Ox parallel to the axis of the impulse, a line Oy drawn outwards from this axis, and a line Oz perpendicular to the plane of the two former. If I and K denote the force- and couple-constituents of the impulse, we have where w denotes the distance of from the axis of the impulse. Since AP—L 2 ^Q, the second and fifth of these equations shew that y = 0, ^=0. Hence ■m is constant throughout the motion, and the remaining quantities are also constant ; in particular PI-LK LIm u = ■(2) AP-L 2 ' AP-L 2 The origin therefore describes a helix about the axis of the impulse, of pitch K P I L' This example is due to Kelvin*. * I.e. ante p. 168. It is there pointed out that a solid of the kind here in question may be constructed by attaching vanes to a sphere, at the middle points of twelve quadrantal arcs drawn so as to divide the surface into octants. The vanes are to be perpendicular to the surface, and are to be inclined at angles of 45° to the respective arcs. Larmor (I.e. ante p. 173) gives another example. "If... we take a regular tetrahedron (or other regular solid), and replace the edges by skew bevel faces placed in such wise that when looked at from any corner they all slope the same way, we have an example of an isotropic helicoid." For some further investigations in the present connection see a paper by Miss Fawcett, "On the Motion of Solids in a Liquid," Quart. Jcnirn. Math. xxvi. 231 (1893). 180 Motion of Solids through a Liquid [chap, vi 131. Before leaving this part of the Subject we remark that the preceding theory applies, with obvious modifications, to the acyclic motion of a liquid occupying a cavity in a mowng solid. If the origin be taken at the centre of inertia of the liquid, the formula for the kinetic energy of the fluid motion is of the type 2T = m (u 2 + v 2 + w 2 ) + P^ 2 + Qq 2 + Rr 2 + 2V'qr + 2Q'rp + ZR'pq. . . .(1) For the kinetic energy is equal to that of the whole fluid mass (m), supposed concentrated at its centre of inertia and moving with this point, together with the kinetic energy of the motion relative to the centre of inertia. The latter part of the energy is easily proved by the method of Arts. 118, 121 to be a homogeneous quadratic function of p, q, r. Hence the fluid may be replaced by a solid of the same mass, having the same centre of inertia, provided the principal axes and moments of inertia be properly assigned. The values of the coefficients in (1), for the case of an ellipsoidal cavity, may be calcu- lated from Art. 110. Thus, if the axes of x, y, z coincide with the principal axes of the ellipsoid, we find P, Q> »-»«?=?. »«£#, *.£=#, P-, * R<=0. Case of a Perforated Solid. 132. If the moving solid have one or more apertures or perforations, so that the space external to it is multiply-connected, the fluid may have a motion independent of that of the solid, viz. a cyclic motion in which the circulations in the several irreducible circuits which can be drawn through the apertures may have any given constant values. We will briefly indicate how the foregoing methods may be adapted to this case. Let k, k , k" , ... be the circulations in the various circuits, and let 8a, 8a', 8a", ... be elements of the corresponding barriers, drawn as in Art. 48. Further, let I, m, n denote the direction-cosines of the normal, drawn towards the fluid at any point of the surface of the solid, or drawn on the positive side at any point of a barrier. The velocity-potential is then of the form </> + #o, where <j> = u<f> 1 + vfc + wfo + pxi + qx* + ^%3, \ ,-^ 0o = KM -r tc co + k co + ... . J The functions cf> 1} cj> 2 , <$>s, %i, %2, %3 are determined by the same conditions as in Art. 118. To determine co, we have the conditions: (1°) that it must satisfy V 2 cw = at all points of the fluid ; (2°) that its derivatives must vanish at infinity; (3°) that dco/dn must = at the surface of the solid; and (4°) that co must be a cyclic function, diminishing by unity whenever the point to which it refers completes a circuit cutting the first barrier once (only) in the positive 131-133] Perforated Solid 181 direction, and recovering its original value whenever the point completes a circuit not cutting this barrier. It appears from Art. 52 that these conditions determine co save as to an additive constant. In like manner the remaining functions co , co", . . . are determined. By the formula (5) of Art. 55, twice the kinetic energy of the fluid is equal to -f»j] (♦ + *>£(♦ + *)*» -|«jJJ i <* + ««fa-^JU;<* + *)&»'- (2) Since the cyclic constants of <£ are zero, and since dfyojdn vanishes at the surface of the solid, we have, by Art. 54 (4), Hence (2) reduces to - P \\^ d S- pK \\^- pK -\\^-- (3) Substituting the values of </>, <£ from (1) we find that the kinetic energy of the fluid is equal to T + K, (4) where T is a homogeneous quadratic function of u, v, w, p, q, r, of the form defined by Art. 121 (2) (3), and 2K=(tc,tc)K 2 + (K', k')k' 2 +... + 2(k,k , )kk' + ..., ...(5) where, for example, (*,*) = -p\\£d<r, > (6) The identity of the different forms of (k, k) follows from Art. 54 (4). Hence the total energy of fluid and solid is given by T = 1& + K, (7) where © is a homogeneous quadratic function of u, v, w, p, q, r of the same form as Art. 121 (8), and K is defined by (5) and (6) above. 133. The 'impulse' of the motion now consists partly of impulsive forces applied to the solid, and partly of impulsive pressures pK, pK, pic", . . . applied uniformly (as explained in Art. 54) over the several membranes which are supposed for a moment to occupy the positions of the barriers. Let us denote by f 1} rji, £i, \ 1} fjL lf v\ the components of the extraneous impulse 182 Motion of Solids through a Liquid [chap, vi applied to the solid. Expressing that the ^-component of the momentum of the solid is equal to the similar component of the total impulse acting on it, we have = fi+ P {[("&+ ••• +PXi + ••• + Ka > + •••) -^dS -ft-S^U^f^J^^ w where, as before, T 2 denotes the kinetic energy of the solid, and T that part of the energy of the fluid which is independent of the cyclic motion. Again, considering the angular momentum of the solid about the axis of x, g- 1 = Xi - p J J (</> + fa) (ny - mz) dS = X 1 -™ + pK \\^ d S + pK '\\ a >' d £dS + (2) Hence, since ® = T + T 1} we have d® f f d<f> fi = du - pK \\ ^dS-oK'Ha'^dS-.. on 1) on ,d<fa .(B) By virtue of Lord Kelvin's extension of Greens Theorem, already referred to, these may be written in the alternative forms (4) Adding to these the terms due to the impulsive pressures applied to the barriers, we have, finally, for the components of the totil impulse of the motion *, X, fJL, V d® d®> d P +x °> ~^ + /X0) dr vo, .(5) where, for example, 6 -^K(i + ^)^ + ^JJ(i + »b)^ + .... \o = ptc \\\7iy-mz+ -p\da + p/c' \\(ny - mz + ^M dor' + .... * Cf. Sir W. Thomson, I.e. ante p. 168 L -(6) 133-134] Components of Impulse 183 It is evident that the constants f , rjo, £o> ^-o, Mo> v o are the components of the impulse of the cyclic fluid motion which would remain if the solid were, by forces applied to it alone, brought to rest. By the argument of Art. 119 ? the total impulse is subject to the same laws as the momentum of a finite dynamical system. Hence the equations of motion of the solid are obtained by substituting from (5) in the equations (1) of Art. 120* 134. As a simple example we may take the case of an annular solid of revolution. If the axis of x coincide with that of the ring, we see by reasoning of the same kind as in Art. 126, 4° that if the situation of the origin on this axis be properly chosen we may write 2T=Au 2 + B(v 2 + iv*) + Pp 2 + Q(q 2 + r 2 ) + ( K , <) k 2 (1) Hence g, rj. £=Au + $ , Bv, Bw; X, /*, v = Pp, Qq, Qr (2) Substituting in the equations of Art. 120, we find dp/dt = 0, or p = const., as is other- wise obvious. Let us suppose that the ring is slightly disturbed from a state of motion in which v, w, p, q, r are zero, i.e. a steady motion parallel to the axis. In the beginning of the disturbed motion v, w, p, q, r will be small quantities whose products we may neglect. The first of the equations referred to then gives du/dt — 0, or u= const., and the remaining equations become .(3) B d J r -(Au + ^)r, Q d J-=- {{A -B)u+£ Q }w, B d f t = (Au + £ )q, Q<^ = {(A-B)u + £ }v. Eliminating r, we find BQ C ^==-(Au + Z ){(A-B)u + £ }v (4) Exactly the same equation is satisfied by w. It is therefore necessary and sufficient for stability that the coefficient of v on the right-hand side of (4) should be negative ; and the time of a small oscillation, when this condition is satisfied, ist 2 r BQ -u () L(Au + £ ){(A-B)u + &}_\ {) We may also notice another case of steady motion of the ring, viz. where the impulse reduces to a couple about a diameter. It is easily seen that the equations of motion are satisfied by £, rj, £, X, /x = 0, and v constant; in which case u — — £JA , r = const. The ring then rotates about an axis in the plane yz parallel to that of 2, at a distance ujr from it J. * This conclusion may be verified by direct calculation from the pressure-formula of Art. 20 ; see Bryan, " Hydrodynamical Proof of the Equations of Motion of a Perforated Solid, ," Phil. Mag. (5) xxxv. 338 (1893). t Sir W. Thomson, I.e. ante p. 168. % For further investigations on this subject we refer to papers by Basset, "On the Motion of a King in an Infinite Liquid," Proc. Camb. Phil. Soc. vi. 47 (1887), and Miss Fawcett, I.e. ante p. 179. 184 Motion of Solids through a Liquid [chap, vi The Forces on a Cylinder moving in Two Dimensions. 134 a. The two-dimensional problem of the motion of a cylindrical body, especially when there is circulation round it, is most simply treated by direct calculation of the pressures on the surface*. We assume as usual that the fluid is at rest at infinity. Taking axes fixed in a cross-section, we denote by (u, v) the velocity of the origin, and by r the angular velocity, the symbols u, v being now required in their original sense as component velocities of the fluid. The pressure- equation is then ;-g-(«-nr)g-(T + »)|-w + «™t (i) where q 2 = u 2 + v 2 . The force (X, Y) and couple (N) to which the pressures on the surface reduce are X = —\plds, Y——\pmds, N = — \p (mx — ly) ds, (2) where I, m are the direction cosines of the normal drawn outwards from an element hs of the contour, and the integration is taken round the perimeter. Now ^ \q 2 lds = — ( u £— + v jr- J dxdy = \(lu + mv) uds, -£ q 2 mds = — M tt — + ^ — J dxdy = \(lu + mv) vds in virtue of the relations dv/dx as du/dy, du/dx + dv/dy — 0. We have here omitted the various line-integrals taken over an infinite enclosing boundary, since at a great distance r the velocity is at most of the order 1/r, whilst 8s is of the order rB0. At the surface of the cylinder we have lu + mv — I (u — ry) + m (v + rx) (4) Hence substituting from (1) in (2) we find X P (3) = — \~ Ids + (mu — Iv) (v + rx) ds — J|*M. + ](* + ») f*A (5) and similarly = -]^mds-](Ti-ry)^ds (6) P Again we find, J q 2 (mx — ly) ds = \(lu + mv) (xv — yu) ds (7) * Aeronautical Research Committee, R. and M. 1218 (1929). For another treatment see Glauert, R. and M. 1215 (1929). 134 a] Forces on a Moving Cylinder 185 Here also, the line-integrals round an infinitely remote boundary are omitted, since we may suppose that at this boundary l/x = m/y, and that lu + mv is of the order 1/r 2 . The formula (2) for N thus becomes — = — \~ (moe — ly) ds + (ux + vy) (Iv — mu) ds = -^(mx-ly)ds-j(jix + vy)^ds (8) We now write, in analogy wifch Arts. 118, 132, <£ = !!(/>! + v0 2 + r;\; + </>o, (9) where (j> represents the circulatory motion which would persist if the cylinder were brought to rest. It is therefore a cyclic function with, say, the cyclic constant k. Comparing with (4) we have, at the surface of the cylinder, dn h d</>2 dn m> d ^=-{mx-ly\ d -p=0 (10) dn J dn 7 In the absence of circulation the energy of the fluid would be T = -ipj(<t>-<f>o) d ^ds (11) Substituting from (9) and (10) this gives 2T = Au 2 + 2Huv + Bv 2 + Rr 2 + 2 (Lu 4- Mv) r, (12) where A=p \lfads, H = p \lfads = p \mfads, "B — p \mfads p]{mx-ly) x ds, p \lxds — p (mx — ly) <$> x ds, M = p\ m^ds = p (mx — ly) <f> 2 ds. The leading terms in (5), (6), and (8) now take the forms --(Au + Hv + Lr) = -^, d /TT „ __ . d dT -^(Hu + Bv + Mr) — g^, .(13) -^(Rr + Lu+Mv) ddT dtdr' .(14) Again, we have p \x ^~ E^ ds = p L ((/)-</) )^ = Hu + Bv + Mr=~ p\y a(0-^ >o) ds ds dT ^(</>-</)o)^ = -(Au + HvH-Lr) = -^ i . (15) 186 Motion of Solids through a Liquid Hence if we write d<f>o [CHAP. VI J-?*-* y-t ds =e> < 16 > the expressions for the forces become X = Y = - ddT dT dt du dv Kpv + par, _ddr dtdv dT r^ + icpu + pPr, dT dT •(17) „ ddT N = -^8r- + V 8H- U 8T-^ aU + ^ v) \ By turning the co-ordinate axes through a suitable angle, the coefficient H can be made to vanish. And by a suitable choice of origin we may also annul the coefficients L, M, or alternatively we may make a = 0, j3 = 0. But these two determinations are in general incompatible, and neither of these special origins can be assumed to coincide with the mean centre of the area of the section. The most interesting case, however, is where the section is symmetrical with respect to each of two perpendicular axes. If these are taken as axes of co-ordinates we have H=0, L = 0, M = 0, a = 0, = 0, (18) and the formulae (17) reduce to X - A -J- + Brv dt dv tcv, .(19) Y = — B t- — Aru + ku, N = -R|-(A-B)uv. To form the equation of motion in this case we have only to modify the inertia coefficients, as in Art. 122. If the distribution of mass is also symmetrical, we write A = A + M, B = B + M, R = R + L, (20) where M represents the mass of the cylinder itself, and L its moment of inertia. Then fin a du „ A Tt -Brv + «pv .(21) B -T- + ^Iru — /cpu = Y, dr L~-(A-B)xw=N, where X y Y, N represent the effect of extraneous forces. When these are absent, and the circulation zero, the solution is as in Art. 127. i34a-i35] Generalized Co-ordinates 187 In the case of a circular section there is no point in supposing the co-ordinate axes to rotate. Putting A — B, r = 0, we have A d f t+Kp u = X, A^-cpn-Y, (22) as in Art. 69. If the section is symmetrical with respect to one axis only, say that of x, we have H = 0, L = 0, B — 0. By a displacement of the origin along the axis of symmetry we can make M = 0,but a will not in general vanish simultaneously. If there is no circulation the new origin corresponds to the c centre of reaction ' of Thomson and Tait*. Equations of Motion in Generalized Co-ordinates. 135. When we have more than one moving solid, or when the fluid is bounded, wholly or in part, by fixed walls, we may have recourse to Lagrange's method of ' generalized co-ordinates.' This was first applied to hydrodynamical problems by Thomson and Tait j*. The systems ordinarily contemplated in Analytical Dynamics are of finite freedom ; i.e. the position of every particle is completely determined when we know the values of a finite number of independent variables or ' generalized co-ordinates' q lt q 2 , ... q n . The kinetic energy T can then be expressed as a quadratic function of the 'generalized velocity components' q lt q 2 , ••• q n . In the Hamiltonian method the actual motion of the system between any two instants t , t-i is compared with a slightly varied motion. If f, rj, f be the Cartesian co-ordinates of any particle m, and X, Y, Z the components of the total force acting on it, it is proved that P l {AT+2(XAf+ YArj + ZAtydt^O, (1) provided the varied motion be such that 2m(%AZ + riAy + JA?) = (2) The summation 2 is understood to include all the particles of the system. The varied motion is usually supposed to be adjusted so that the initial and final positions of each particle shall be respectively the same as in the actual motion. The quantities Af, At;, Af then vanish at each limit of integration, and the condition (2) is fulfilled. For a conservative system free from extraneous force (1) takes the form A \ t \T-V)dt = (3) * Natural Philosophy, Art. 321. t Ibid. Art. 331, 188 Motion of Solids through a Liquid [chap, vi In words, if the actual motion of the system between any two configura- tions through which it passes be compared with any slightly varied motion, between the same configurations, which the system is (by the application of suitable forces) made to execute in the same time, the time-integral of the 'kinetic potential '* V— T is, stationary. In terms of generalized co-ordinates, the equation (1) takes the form \ t \&T+Q 1 Aq 1 + Q 2 &q 2 +... + Q n &q n )dt = 0, (4) from which Lagrange's equations d dT dT _ ~ ,p,. dtdir" d^ r ~ Vr W can be deduced by a known process. 136. Proceeding now to the hydrodynamical problem, let q lt q 2 , ... q n be a system of generalized co-ordinates which serve to specify the configuration of the solids. We will suppose, for the present, that the motion of the fluid is entirely due to that of the solids, and is therefore irrotational and acyclic. In this case the velocity-potential at any instant will be of the form <£ = 9i0i + ?202+ ... +q n <t>n> (1) where </>i, <j> 2 , ... are determined in a manner analogous to that of Art. 118. The formula for the kinetic energy of the fluid is then 2T = - /J Jj</,^^=A 11 gi 2 + A 22 g 2 2 +...+2A 12 ^ 2 +..., (2) Kr=~p J J* r ^ dS, A rs = -p j|</> r g dS = -p J J*. ^ dS, . . .(3) where the integrations extending over the instantaneous positions of the bounding surfaces of the fluid. The identity of the two forms of A rs follows from Green's Theorem. The coefficients A rr , A rs will in general be functions of the co-ordinates q 1} q 2 , ... q n . If we add to (2) twice the kinetic energy, Ti, of the solids themselves, we get an expression of the same form, with altered coefficients, say 2T = Autf + A 22 q 2 * + ... +2A u q 1 q 2 + (4) It remains to shew that, although our system is one of infinite freedom, the equations of motion of the solids can, under the circumstances pre- supposed, be obtained by substituting this value of T in the Lagrangian equations, Art. 135 (5). We are not at liberty to assume this without further examination, for the positions of the various particles of the fluid are * The ndime was introduced by Helmholtz, "Die physikalische Bedeutung des Princips der kleinsten Wirkung," Crelle, c. 137, 213 (1886) [Wiss. Abh. iii. 203]. 135-136] Application to Hydrodynamics 189 not determined by the instantaneous values q lf q 2 , ... q n of the co-ordinates of the solids. For instance, if the solids, after performing various evolutions, return each to its original position, the individual particles of the fluid will in general be found to be finitely displaced*. Going back to the general formula (1) of Art. 135, let us suppose that in the varied motion, to which the symbol A refers, the solids undergo no change of size or shape, and that the fluid remains incompressible, and has, at the boundaries, the same displacement in the direction of the normal as the solids with which it is in contact. It is known that under these conditions the terms due to the internal reactions of the solids will disappear from the sum The terms due to the mutual pressures of the fluid elements are equivalent to or lip (IA% + mA v + nA£) dS + [lip (^ + ^ + d -^\ dxdydz, where the former integral extends over the bounding surfaces, and I, m, n denote the direction-cosines of the normal, drawn towards the fluid. The volume-integral vanishes by the condition of incompressibility ¥ f + T ? + X ? -0 < 5 > ox ay dz The surface-integral vanishes at a fixed boundary, where ZAf + mAr) + nA£ = 0; and in the case of a moving solid it is cancelled by the terms due to the pressure exerted by the fluid on the solid. Hence the symbols X, Y, Z may be taken to refer only to the remaining forces acting on the system, and we may write Z(XA£+YA v + ZAO=QiA qi + Q 2 Aq 2 + ... + Q n Aq n , (6) where Q 1} Q 2} ... Q n are generalized components of force. The varied motion of the fluid has sfcill a high degree of generality. We will now further limit it by supposing that while the solids are, by suitable forces applied to them, made to execute an arbitrary motion, the fluid is left to take its own course in consequence of this. The varied motion of the fluid may accordingly be taken to be irrotational, in which case the varied kinetic energy T+AT of the system will be the same function of the varied co-ordinates q r + Aq r , and the varied velocities q r + Aq r , that the actual energy T is of q r and q r . * As a simple example, take the case of a circular disk which is made to move, without rotation, so that its centre describes a rectangle two of whose sides are normal to its plane ; and examine the displacements of a particle initially in contact with the disk at its centre. .7)1 190 Motion of Solids through a Liquid [chap, vi Again, considering the particles of the fluid alone, we shall have, on the same supposition, 2m (£Af + v^v + SAf) = - P jjj(g A* + g Ay + g A*) fofy* = p [(<£ (ZAf + rnA?? + wAf) <Z£, where use has again been made of the condition (5) of incompressibility. By the kinematical condition to be satisfied at the boundaries, we have and therefore = (Aii^ + A 12 q 2 + • • • + A ln q n ) Aq x + (Aaji + A2252 + . ■ • + A 2?l ^n) Ag 2 -f ... +(A n i^ 1 + A n2 J2+ ... +A nn ^ n )A^ n -s* Aft t5i^ + - + "8S;^ (7) by (1), (2), (3) above. If we add the terms due to the solids, we find that the condition (2) of Art. 135 still holds; and the deduction of Lagrange's equations then proceeds in the usual manner. 137. As a first application of the foregoing theory we may take an example given by Thomson and Tait*, where a sphere is supposed to move in a liquid which is limited only by an infinite plane wall. Taking, for simplicity, the case where the centre moves in a plane perpendicular to that of the wall, let us specify its position at time t by rectangular co-ordinates x, y in this plane, of which y denotes distance from the wall. We have 2T=Ax 2 + By 2 , (1) where A and B are functions of y only, it being plain that the term xy cannot occur, since the energy must remain unaltered when the sign of x is reversed. The values of A , B can be written down from the results of Arts. 98, 99, viz. if m denote the mass of the sphere, and a its radius, we have J^«+|,rpa*(l + AJjJ), *«m+i«r^(r + t|£), (2) approximately, if y be great in comparison with a. The equations of motion give >>=* >>-i(f* 2+ S^H (3) where X, Y are the components of extraneous force, supposed to act on the sphere in a line through the centre. * Natural Philosophy , Art. 321. 136-138] Application to Hydrodynamics 191 If there be no extraneous force, and if the sphere be projected in a direction normal to the wall, we have x=0, and By 2 =const (4) Since B diminishes as y increases, the sphere experiences an acceleration from the wall. Again, if the sphere be constrained to move in a line parallel to the wall, we have y = 0, and the necessary constraining force is y =-if* 2 - < 5 > Since dAjdy is negative, the sphere appears to be attracted by the wall. The reason of this is easily seen by reducing the problem to one of steady motion. The fluid velocity will evidently be greater, and the pressure therefore less, on the side of the sphere next the wall than on the further side ; see Art. 23. The above investigation will also apply to the case of two spheres projected in an unlimited mass of fluid, in such a way that the plane y==0 is a plane of symmetry in all respects. 138. Let us next take the case of two spheres moving in the line of centres. The kinematical part of this problem has been treated in Art. 98. If we now denote by x, y the distances of the centres of the spheres A, B from some fixed origin in the line joining them, we have 2T=Lx 2 -2Mxy + JYy\ (1) where the coefficients Z, M t N are functions of y — #, or c, the distance between the centres. Hence the equations of motion are d . ,,. ,,.. . (dL .„ a dM . . dN .A „ where X, Y are the forces acting on the spheres along the line of centres. If the radii a, b are both small compared with c, we have, by Art. 98 (15), keeping only the most important terms, L=m + lTrpd\ M^Zirp — , iV=m' + f 7rp& 3 , (3) approximately, where m, m' are the masses of the two spheres. Hence to this order of approximation dL n dM a?b* dN „. ^ = ' ^ = - 6 ^^-' -aTc =0 - If each sphere be constrained to move with constant velocity, the force which must be applied to A to maintain its motion is This tends towards B, and depends only on the velocity of B. The spheres therefore appear to repel one another ; and it is to be noticed that the apparent forces are not equal and opposite unless x=±y. Again, if each sphere make small periodic oscillations about a mean position, the period being the same for each, the mean values of the first terms in (2) will be zero, and the spheres therefore will appear to act on one another with forces equal to top^g-m -(5) ■(2) 192 Motion of Solids through a Liquid [chap, vi where [xy] denotes the mean value of xy. If x, y differ in phase by less than a quarter- period, this force is one of repulsion, if by more than a quarter-period it is one of attraction. Next, let B perform small periodic oscillations, while A is held at rest. The mean force which must be applied to A to prevent it from moving is X=ilw («) where [y 2 ] denotes the mean square of the velocity of B. To the above order of approxi- mation dN/dc is zero ; on reference to Art. 98 we find that the most important term in it is — 127rpa 3 6 6 /c 7 , so that the force exerted on A is attractive, and equal to ^p-fm (7) This result comes under a general principle enunciated by Kelvin. If we have two bodies immersed in a fluid, one of which {A ) performs small vibrations while the other (B) is held at rest, the fluid velocity at the surface of B will on the whole be greater on the side nearer A than on that which is more remote. Hence the average pressure on the former side will be less than that on the latter, so that B will experience on the whole an attraction towards A. As practical illustrations of this principle we may cite the apparent attraction of a delicately-suspended card by a vibrating tuning-fork, and other similar phenomena studied experimentally by Guthrie* and explained in the above manner by Kelvin t. Modification of Lagrange's Equations in the case of Cyclic Motion. 139. We return to the investigation of Art. 136, with the view of adapting it to the case where the fluid has cyclic irrotational motion through channels in the moving solids, or (it may be) in an enclosing vessel, in- dependently of the motion due to the solids themselves. Let us imagine barrier-surfaces to be drawn across the several apertures. In the case of channels in a containing vessel we shall suppose these ideal surfaces to be fixed in space, and in the case of channels in a moving solid we shall suppose them to be fixed relatively to the solid. Let ^, #', % ', ... be the fluxes at time t across, and relative to, the several barriers ; and let X> X> X> •" De * ne time-integrals of these fluxes, reckoned from some arbitrary epoch, these quantities determining (therefore) the volumes of fluid which have up to the time t crossed the respective barriers. It will appear that the analogy with a dynamical system of finite freedom is still conserved, provided the quantities %, %', %", ... be reckoned as generalized co-ordinates of the system, in addition to those (q x , q 2 , ... q n ) which specify the positions of the moving solids. It is obvious already that the absolute values of x> X> X> ••• w ^ not en t er into the expression for the kinetic energy, but only their rates of variation. In the first place, we may shew thai; the motion of the fluid, in any given configuration of the solids, is completely determined by the instantaneous * "On Approach caused by Vibration," Phil. Mag. (4) xl. 345 (1870). t Reprint of Papers on Electrostatics, dbc. Art. 741. For references to further investigations, both experimental and theoretical, by C. A. Bjerknes and others, on the mutual influence of oscillating spheres in a fluid, see Hicks, "Eeport on Kecent Kesearcb.es in Hydrodynamics," Brit. Ass. Rep. 1882, pp. 52...; Love, Encycl. d. math. Wiss. iv. (3), pp. Ill, 112. 138-139] Hamiltonian Method 193 values of q lf q 2 , • •• q n > % X '» X> -^ or ^ there were two modes of irrotational motion consistent with these values, then, in the motion which is the difference of these, the boundaries of the fluid would be at rest, and the flux across each barrier would be zero. The formula (5) of Art. 55 shews that under these conditions the kinetic energy would vanish. It follows that the velocity-potential can be expressed in the form </> - $i*i + q*<t>2+ ... +4n<l>n + xto+x' n '+ W Here cb r is the velocity-potential of a motion in which q r alone varies and the flux across each barrier is accordingly zero. Again O is the velocity- potential of a motion in which the solids are all at rest, whilst the flux through the first aperture is unity, and that through every other aperture is zero. It is to be observed that <f> 1} <£ 2 , ••• </> n > &, &', ••• are in general all of them cyclic functions, which may however be treated as single-valued, on the conventions of Art. 50. The kinetic energy of the fluid is given by the expression 2T '///{©'♦S)'*®}*** < 2 > where the integral is taken over the region occupied by the fluid at the instant under consideration. If we substitute from (1) we obtain T as a homogeneous quadratic function of q 1} q 2 , ... q n , %, %', x > ••• w ^ n coefficients which depend on the instantaneous configuration of the solids, and are there- fore functions of q 1} q 2 , ... q n only. Moreover, we find, by Art. 53 (1), where k, k, ... are the cyclic constants of (/>, and the first surface -integral is to be taken over the surfaces of the solids, and the remaining ones over the several barriers. By the conditions which determine 12, this reduces to the first equation of the system : dT dT , %-'*' ty= pK > (3) These shew that p/c, pre', . . . are to be regarded as the generalized components of momentum corresponding to the velocity-components %, ^', ..., respec- tively. We have recourse to the general Hamiltonian formula (1) of Art. 135. We will suppose that the varied motion of the solids is subject only to the condition that the initial and final configurations are to be the same as in the actual motion ; also that the initial position of each particle of the fluid is the same in the two motions. The expression 2m(fAf + i/Ai7 + £A£) ..(4) 194 Motion of Solids through a Liquid [chap, vi will accordingly vanish at time t , but not in general at time t ly in the absence of further restrictions. We will now suppose that the varied motion of the fluid is irrotational, and accordingly determined by the instantaneous values of the varied generalized co-ordinates and velocities. Considering the particles of the fluid alone, we have = p \\<l>(lAg+mAii+nA£)dS+pK [|(ZAf + mAr) + nA£) do- + P k' [l(lA£ + mA v +nAZ)d<r' + ..., (5) where I, m, n are the direction-cosines of the normal to an element of the bouuding surface, drawn towards the fluid, or (as the case may be) of the normal to an element of a barrier, drawn in the direction in which the corresponding circulation is estimated. At time t± we shall have ZAf + mA77 + ?iAf=0 at the surface of the solids, as well as at the fixed boundaries. Again, if A B represent one of the barriers in its position at time t ly and if A f B' represent the locus at the same instant, in the varied motion, of those particles which in the actual motion occupy the position AB, the volume included between AB and A'B' will be equal to the corre- sponding A%, whence .(6) (ZA£ + mA?7 4- nA? ) da = A%, [ [ (ZAf -I- m&y + **Af ) da' = A%', The varied circulations are, from instant to instant, still at our disposal. We may suppose them to be so adjusted as to make A%, A^', ... vanish at time ti. The expression (4) will accordingly vanish, and if we further suppose that the external forces do on the whole no work when the boundary of the fluid is at rest, whatever relative displacements be given to the parts of the fluid, we have f' , {AT+Q 1 A ?1 + Q 2 Ag 2 + ...+Q n A^)^ = 0, (7) s before. By a partial integration, and remembering that by hypothesis Aq ly Aq 2 , ... A^, A % , A*', ... 139-140] Ignoration of Co-ordinates 195 vanish at the limits t ,t ly but are otherwise independent, we obtain n equations of the type dtdq r dq r ~^ W togetherwith ||f = > S |f " °« (9) 140. Equations of the type (8) and (9) present themselves in various problems of ordinary Dynamics, e.g. in questions relating to gyrostats, where the co-ordinates %, %', . .., whose absolute values do not affect the kinetic or the potential energy of the system, are the angular co-ordinates of the gyrostats relative to their frames. The general theory of such systems has been treated by Routh*, Thomson and Taitf, and other writers. We have seen that -^-.—pK, ~-r, = pi<! y . . . , (10) and the integration of (9) shews that the quantities k, k, ... are constants with regard to the time, as is otherwise known (Art. 50). Let us write R = T-p K x-p«'x- (11) The equations (10), when written in full, determine %, %', ... as linear functions of k, k, ... and q ly q 2y ... q n \ and by substitution in (11) we can express R as a homogeneous quadratic function of the same quantities, with coefficients which of course in general involve the co-ordinates q ly q 2y ... q n . On this supposition we have, performing an arbitrary variation & on both sides of (11), and omitting terms which cancel by (10), dR ^ dR * ' dR 5, =-r dgi + . . . + ^— dtfi + . . . + ^— OK + . . . = ||Sg 1 + ... + ||85 1+ ...-^8*- (12) where, for brevity, only one term of each kind is exhibited. Hence we obtain 2n equations of the types dR = dT dR^dT dq r dq r ' dq r dq r ' togetherwith _ = - p ^, — , = -px', (14) Hence the equations (8) may be written d dR dR _ ^ ,,.v dt dq r dq r * On the Stability of a Given State of Motion (Adams Prize Essay), London, 1877; Advanced Rigid Dynamics, 6th ed., London, 1905. f Natural Philosophy, 2nd ed., Art. 319 (1879). See also Helmholtz, "Principien der Statik monocyclischer Systeme," Crelle, xcvii. (1884) [Wiss. Abh. iii. 179]; Larmor, "On the Direct Application of the Principle of Least Action to the Dynamics of Solid and Fluid Systems," Proc. Lond. Math. Soc. (1) xv. (1884) [Papers, i. 31]; Basset, Proc. Camb. Phil. Soc. vi. 117 (1889). 196 Motion of Solids through a Liquid [chap, vi where the velocities %, %', ... corresponding to the 'ignored' co-ordinates %, %', ... have now been eliminated*. 141. In order to shew more explicitly the nature of the modification introduced by the cyclic motions into the dynamical equations, we proceed as follows. If we substitute in (11) from (14), we obtain y =*-("g + *'g + -) < 16 > Now, remembering the composition of R, we may write for a moment R — ^2,0 + R\,l + ^0,2 j (17) where R 2y o is a homogeneous quadratic function of q x , q%, ... q n , i2 0)2 is a homogeneous quadratic function of k, k\ ... , and R lf i is bilinear in these two sets of variables. Hence (16) takes the form T — i?2,0 — ^0,2 > (18) or, as we shall henceforth write it, t=® + k, (19) where tlT and K are homogeneous quadratic functions of q lt q%, ... q n , and of k, k, ... , respectively. It follows also from (17) that r = i&-r- fab- fab- ...- p«in, (20) where /3i , /?2 > • • • are linear functions of k, k, ... , say /3 1 = a 1 K + a 1 'tc' + ... , * /3„ = a w /c + a w V + The meaning of the coefficients a (in the hydrodynamical problem) appears from (14) and (20). We find .(22) . dK . PX = d/c + a i?i + «2<?2 + ... 4- a n ?r ., dK ,. px = ^7 + «igi + «2?2+... +« w g r » which shew that a r is the contribution to the flux of matter across the first barrier due to unit rate of variation of the co-ordinate q ry and so on. If we now substitute from (20) in the equations (15) we obtain the general equations of motion of a ' gyrostatic system,' in the formf * This investigation is due to Bouth, I.e.; cf. Whittaker, Analytical Dynamics, Art. 38. t These equations were first given in a paper by Sir W. Thomson, "On the Motion of Kigid Solids in a Liquid circulating irrotationally through perforations in them or in a Fixed Solid," Phil. Mag. (4) xlv. 332 (1873) [Papers, iv. 101]. See also C. Neumann, Hydrodynamische Untersuchungen (1883). ho-142] Kineto- Statics 197 lf-| + o.«)* + a.8)* + .. r+ a»)*. + |-ft. (23) where C'.')-fr-t <"> It is important to notice that (r, s) = — (5, r), and (r, r) = 0. If in the equations of motion of a fully-specified system of finite freedom (Art. 135 (4)) we reverse the sign of the time-element Bt, the equations are unaltered. The motion is therefore reversible ; that is to say, if as the system is passing through any assigned configuration the velocities qi , q 2 , • . • q n be all reversed, it will (if the forces be always the same in the same configuration) retrace its former path. It is important to observe that this statement does not in general hold of a gyros tatic system ; thus, the terms in (23) which are linear in q lt q 2 , ... q n change sign with 8t, whilst the others do not. Hence, in the present application, the motion of the solids is not reversible, unless indeed we imagine the circulations tc, k, ... to be reversed simultaneously with the velocities q lf q 2 , ... q n *• If we multiply the equations (23) by q lt q 2 , ... q n in order, and add, we find, by an easy adaptation of the usual process, g(« + JO = &$! + && +... + &$«, (25) or, if the system be conservative, *& + !{+¥= const (26) 142. The results of Art. 141 may be applied to find the conditions of equilibrium of a system of solids surrounded by a liquid in cyclic motion. This problem of ' Kineto-Statics/ as it may be termed, is however more naturally treated by a simpler process. The value of <f> under the present circumstances can be expressed in the alternative forms #-xP+#d'.+ ..., (i) (j) = fCCO + K(o' + . . . ; . (2) and the kinetic energy can accordingly be obtained as a homogeneous quad- ratic function either of %%> ... , or of k, k, ... , with coefficients which are in each case functions of the co-ordinates q 1} q 2 , ... q n which specify the * Just as the motion of the axis of a top cannot be reversed unless we reverse the spin. 198 Motion of Solids through a Liquid [chap, vi configuration of the solids. These two expressions for the energy may be distinguished by the symbols T and K, respectively. Again, by Art. 55 (5) we have a third formula 2T= P kx + PI c'x' + (3) The investigation at the beginning of Art. 139, shortened by the omission of the terms involving q ly q 2 , ... q n , shews that dT , dT P«=g£. P"=W' (4) Again, the explicit formula for K is = (*, K)K 2 + (fc', fc')K' 2 +...+2(/C ) K')fCK'+..., (5) where = (tc, k) k + (/c, k) k + . . . = - p Lp do: and so on. Hence dK die We thus obtain p % = — ) p tf = — , (7) Again, writing T + K for IT in (3), and performing a total variation 8 on both sides of the resulting identity, we find, on omitting terms which cancel in virtue of (4) and (7)*, ^ + ^=0 (8) dq r dq r This completes the requisite analytical formulae f. If we now imagine the solids to be guided from rest in the configuration (<2i, <}2> ••• <?n) to rest in an adjacent configuration (#i + Aq 1} q 2 + Aq 2 , ... q n + Aq n ), the work required is Qi&qi + Qi^q 2 4- ... + Q n &q n , where Q iy Q 2 , •• Q n are the components of extraneous force which have to be applied to neutralize the pressures of the fluid on the solids. This must be equal to the increment AK of the kinetic energy, calculated on the supposition that the circulations k, tc', ... are constant. Hence <2'=af W * It would be sufficient to assume either (4) or (7) ; the process then leads to an independent proof of the other set of formulae. t It may be noted that the function R of Art. 140 now reduces to - K. 142-143] Kineto-Statics 199 The forces representing the pressures of the fluid on the solids (when these are held at rest) are obtained by reversing the signs, viz. they are given by *— .£; (10) the solids therefore tend to move so that the kinetic energy of the cyclic motion diminishes. In virtue of (8) we have, also, Q/ = ^° • (11) dq r 143. The formula (19) of Art. 141 may be applied to find approximate expressions for the forces on a solid immersed in a non-uniform stream*. Suppose we have a solid maintained at rest in a cyclic region in which a fluid is circulating irrotationally, and let K be the energy of the fluid, which will of course vary with the position of the solid. We will suppose the dimensions of the latter to be so small compared with the distances from the walls of the region that its position may be sufficiently given by point- co-ordinates (x, y, z). We have, then, for the components of the force exerted on it by the pressures of the fluid, X= _3Z Y _ZK Z = JK ox dy cz It remains to find, approximately, the form of this function K of oc, y, z. Let (u, v, w) be the velocity which the fluid would have at (x, y, z) if the solid were absent. If the solid were made to move with this velocity, and were of the same density as the surrounding fluid, the energy would be approximately the same as if the whole were fluid. It follows from Art. 141 (19) that in this case the energy of the fluid would be ® + K, where 2^ = Au 2 + Bv 2 + Cw 2 + 2A'vw + 2B' wu + 2C'uv, (2) by Art. 124, and that of the solid would be i p Q(u 2 + v 2 + w 2 ), (3) where Q is the volume displaced. The expression r ® + ipQ(u 2 + v 2 + w 2 ) + K (4) has therefore a constant value, viz. that of the energy of a fluid filling the region, and having the given circulations. This determines the form of K. Hence Y= ^ +i ^| (M2+w2+w2) ^ - (5) Z-.^ + ipQ^(u« + «• + «»). * G. I. Taylor, " The Forces on a Body placed in a Curved or Converging Stream of Fluid," Proc. Roy. Soc. cxx. 260 (1928). 200 Motion of Solids through a Liquid [chap, vi Since the forces on the solid must depend only on the motion of the fluid in the immediate neighbourhood, these expressions are general, and inde- pendent of the special conception employed in their derivation. If the direction of the undisturbed stream, near the solid, be taken as the axis of x, the results simplify. Putting v = 0, w=0, we have /a ^3m _. dw ~,,dv) T={ ( A +P <; + B<g + o<! If, further, the stream is symmetrical with respect to the planes y=0, 2=0 we have ' = 0, dujdz=0, and therefore also dv/dx = 0, dw/dx=0, on account of the assumed irrotational character. The symmetry also requires dw/dy=dv/dz=0. Hence -{ .(6) •(7) First suppose that one of the axes of permanent translation (Art. 124) coincides with the direction of the stream. Then C = 0, B' = 0, and X=(A+p£)/, Y=0, Z=0, (8) where / is the acceleration in the undisturbed stream. Thus if the solid is spherical, A = §7rpa 3 , Q=^vra 3 , X = 27rpa 3 /. For a circular cylinder, reckoning per unit length, A = 7rpa 2 , Q = 7rpa 2 , X = Zirpa 2 f. Next suppose merely that two of the axes of permanent translation lie in a plane with the direction of the stream. If the plane in question be that of xy we have A' = 0, B' = 0. If the stream is symmetrical about the axis of x, we have, further, dv _ aw__ 1 du dy ~ dz 2 dx' and the forces reduce to X = (A+ P G)/, Y=4C'/, Z=0 (9) In the case of a circular disk, A = |pa 3 cos 2 a, C'= -| pa 3 sin a cos a, $=0, where a is the angle which the stream makes with the axis of symmetry. In the two- dimensional case of the elliptic cylinder, A = n-p (b 2 cos 2 a + a 2 sin 2 a), C = irp (a 2 — b 2 ) sin a cos a, Q = nab, where a is now the inclination of the stream to the major axis*. The above theory has an interest in connection with the ' pressure-drop ' in a wind- channel, as used for measuring the drag of aircraft models. The stream of air converges slightly towards the fan at the forward end of the tunnel, and the increase of velocity implies a fall of pressure. We have then »/=-! oo) * These particular cases have been verified by direct calculation of the effect of the fluid pressures: Aeronautical Research Committee, R. and M. 1164 (1928). 143-144] Kineto- Statics 201 The preceding formulae shew that it would be incorrect to calculate the value of X from the observed pressure-gradient as if it were a statical question, in which case we should have "X. = pQf simply*. Some further interesting examples of Kineto-Statics (not reproduced in the present edition) have been discussed by Sir W. Thomson f, Kirchhoff {, and Boltzmann§. 144. We here take leave of this branch of our subject. To avoid, as far as may be, the suspicion of vagueness which sometimes attaches to the use of ' generalized co-ordinates,' an attempt has been made in this Chapter to put the question on as definite a basis as possible, even at the expense of some degree of prolixity in the methods. To some writers || the matter has presented itself as a much simpler one. The problems are brought at one stroke under the sway of the ordinary formulae of Dynamics by the imagined introduction of an infinite number of 'ignored co-ordinates,' which would specify the configuration of the various particles of the fluid. The corresponding components of momentum are assumed all to vanish, with the exception (in the case of a cyclic region) of those which are represented by the circulations through the several apertures. From a physical point of view it is difficult to refuse assent to such a generalization, especially when it has formed the starting-point of all the development of this part of the subject; but it is at least legitimate, and from the hydrodynamical standpoint even desirable, that it should be verified a posteriori by independent, if more pedestrian, methods. Whichever procedure be accepted, the result is that the systems con- templated in this Chapter are found to comport themselves (so far as the 'palpable' co-ordinates q 1} q 2 , ... q n are concerned) exactly like ordinary systems of finite freedom. The further development of the general theory belongs to Analytical Dynamics, and must accordingly be sought for in books and memoirs devoted to that subject. It may be worth while, however, to remark that the hydrodynamical systems afford extremely interesting and beautiful illustrations of the Principle of Least Action, the Reciprocal Theorems of Helmholtz, and other general dynamical theories. * G. I. Taylor, I.e. t "On the Forces experienced by Solids immersed in a Moving Liquid," Proc. R. S. Edin. 1870 [Reprint, Art. xli.]. % I.e. ante p. 54. § "Ueber die Druckkrafte welche auf Einge wirksam sind die in eine bewegte Flussigkeit tauchen," Crelle, lxxiii. (1871) [Wiss. Abh. i. 200]. || See Thomson and Tait, and Larmor, 11. cit. ante p. 195. CHAPTER VII VORTEX MOTION 145. Our investigations have thus far been confined for the most part to the case of irrotational motion. We now proceed to the study of rotational or 'vortex' motion. This subject was first investigated by Helmholtz*; other and simpler proofs of some of his theorems were afterwards given by Kelvin in the paper on vortex motion already cited in Chapter ill. We shall, throughout this Chapter, use the symbols f, 77, f to denote, as in Chapter III., the components of vorticity, viz. <._dw dv _du dw ^__dv _du . dy dz* dz dx' dx dy A line drawn from point to point so that its direction is everywhere that of the instantaneous axis of rotation of the fluid is called a 'vortex-line.' The differential equations of the system of vortex-lines are dx dy _ dz , g v T = 7 _ 7 () If through every point of a small closed curve we draw the corresponding vortex-line, we mark out a tube, which we call a ' vortex-tube.' The fluid contained within such a tube constitutes what is called a ' vortex-filament,' or simply a ' vortex.' Let ABC, A'B'C be any two circuits drawn on the surface of a vortex- tube and embracing it, and let A A' be a connecting line also drawn on the surface. Let us apply the theorem of Art. 32 to the circuit ABGAA'G'B'A A and the part of the surface of the tube bounded by it. Since /f +ra77 + nf=0 at every point of this surface, the line-integral j(udx + vdy + wdz), taken round the circuit, must vanish ; i.e. in the notation of Art. 31 I(ABCA) + I(AA') + I(A'C'B , A') + I(A'A) = 0, which reduces to I (ABC A) = 1 {A'B'C A'). Hence the circulation is the same in all circuits embracing the same vortex- tube. * "Ueber Integrate der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen," Crelle, lv. (1858) [Wiss. Abh. i. 101]. 145-146] Persistence of Vortices 203 Again, it appears from Art. 31 that the circulation round the boundary of any cross-section of the tube, made normal to its length, is coa, where <o, = (f 2 + 7? 2 + £ 2 )^, is the resultant vorticity of the fluid, and a the infinitely small area of the section. Combining these results we see that the product of the vorticity into the cross-section is the same at all points of a vortex. This product is conveniently taken as a measure of the ' strength ' of the vortex*. The foregoing proof is due to Kelvin ; the theorem itself was first given by Helmholtz, as a deduction from the relation which follows at once from the values of £, rj, J" given by (1). In fact writing, in Art. 42 (1), f, y, £ for U, V, W, respectively, we find fJ(K'+m + Od8~o, (4) where the integration extends over any closed surface lying wholly in the fluid. Applying this to the closed surface formed by two cross-sections of a vortex-tube and the part of the walls intercepted between them, we find ft>i0i = 0)20*2, where ©i, co 2 denote the vorticities at the sections oi, cr 2 , respectively. Kelvin's proof shews that the theorem is true even when f, 77, f are discontinuous (in which case there may be an abrupt bend at some point of a vortex), provided only that u, v, w are continuous. An important consequence of the above theorem is that a vortex-line cannot begin or end at any point in the interior of the fluid. Any vortex- lines which exist must either form closed curves, or else traverse the fluid, beginning and ending on its boundaries. Compare Art. 36. The theorem of Art. 32 (3) may now be enunciated as follows : The circulation in any circuit is equal to the sum of the strengths of all the vortices which it embraces. 146. It was proved in Art. 33 that in a perfect fluid whose density is either uniform or a function of the pressure only, and which is subject to forces having a single-valued potential, the circulation in any circuit moving with the fluid is constant. Applying this theorem to a circuit embracing a vortex-tube we find that the strength of any vortex is constant. If we take at any instant a surface composed wholly of vortex-lines, the circulation in any circuit drawn on it is zero, by Art. 32, for we have 1% + my + ft? = at every point of the surface. The preceding Art. shews that if the surface be now supposed to move with the fluid, the circulation will always be zero in any circuit drawn on it, and therefore the surface will * The circulation round a vortex being the most natural measure of its intensity. 2(M Vortex Motion [chap, vii always consist of vortex-lines. Again, considering two such surfaces, it is plain that their intersection must always be a vortex-line, whence we derive the theorem that the vortex-lines move with the fluid. This remarkable theorem was first given by Helmholtz for the case of incompressibility ; the preceding proof, by Kelvin, shews that it holds for all fluids subject to the conditions above stated. The theorem that the circulation in any circuit moving with the fluid is invariable constitutes the sole and sufficient appeal to Dynamics which it is necessary to make in the investigations of this Chapter. It is based on the hypothesis of a continuous distribution of pressure, and (conversely) implies this. For if in any problem we have discovered functions u, v, w of x, y, z, t which satisfy the kinematical conditions, then, if this solution is to be also dynamically possible, the relation between the pressures about two moving particles A, B must be given by the formula (2) of Art. 33, viz. *£ + Cl-itf\ B = -^- \ B (udx + vdy + ivdz) (1) p \a ■Lftjj It is therefore necessary and sufficient that the expression on the right-hand side should be the same for all paths of integration (moving with the fluid) which can be drawn from A to B. This is secured if, and only if, the assumed values of u, v, w make the vortex-lines move with the fluid, and also make the strength of every vortex constant with respect to the time. It is easily seen that the argument is in no way impaired if the assumed values of u, v, w make f, rj, f discontinuous at certain surfaces, provided only that u, v, w are themselves everywhere continuous. On account of their historical interest, one or two independent proofs of the preceding theorems may be briefly indicated, and their mutual relations pointed out. Of these proofs, perhaps the most conclusive is based upon a slight generalization of some equations given originally by Cauchy in the introduction to his great memoir on Waves * and employed by him to demonstrate Lagrange's velocity-potential theorem. The equations (2) of Art. 15 yield, on elimination of the function % by cross-differentia- tion, du dx du dx dv dy dv dy dw dz dw dz dw dv db dc dc db db dc dc 36 cb dc dc db ~ 86 dc (where u, v, w have been written in place of dx/dt, dy/ot, dz/dt, respectively), with two symmetrical equations. If in these equations we replace the differential coefficients of u, v, w with respect to a, 6, c, by their values in terms of differential coefficients of the same quantities with respect to x, y, z, we obtain > d(y> g ) , %{z,x) / . d(x,y) _ ■) ? 8(6,c) +?7 a(6,c) i "^ 9(6,c) *° ^d(c,a)' i ' V d(c,a)' f ' i d(c,a) ? ' ? 3(a,6)' r ^a(a,6)" + " J, 8(a, 6) i0 ' * I.e. ante p. 17. 146] Helmholtz' Equations 205 If we multiply these by dx/da, dx/db, dx/dc, in order, and add, then, taking account of the Lagrangian equation of continuity (Art. 14 (1)) we deduce the first of the following three symmetrical equations : £ ^^, Vo dx . Co dx p p da p db p dc p da p db p dc ,(3) p p da p 96 p dc In the particular case of an incompressible fluid (p=p ) these differ only in the use of the notation £, rj, £ from the equations given by Cauchy. They shew at once that if the initial values £ , ?7 , £ of the component vorticities vanish for any particle of the fluid, then £, 77, £ are always zero for that particle. This constitutes in fact Cauchy's proof of Lagrange's theorem. To interpret (3) in the general case, let us take at time t = 0& linear element coincident with a vortex-line, say da, 8b, fc«c&, - m - Co Po Po e — . Po' where e is infinitesimal. If we suppose this element to move with the fluid the equations (3) shew that its projections on the co-ordinate axes at any other time will be given by 8x, By. 8z=e-, e-, e-, * P P P i.e. the element will still form part of a vortex-line, and its length (8s, say) will vary as ca/p, where o> is the resultant vorticity. But if a be the cross-section of a vortex -filament having 8s as axis, the product p<r8s is constant with regard to the time. Hence the strength oacr of the vortex is constant*. The proof given originally by Helmholtz depends on a system of three equations which, when generalized so as to apply to any fluid in which p is a function of p only, become t £ du rj du £ du pdx pdy pdz Dt\p) pfa + p*-' r ~ " D fC Dt\p) _i?2f! + 2: Dt \p) pdx p dy pdz p dz ' .(4) These may be obtained as follows. The dynamical equations of Art. 6 may be written, when a force-potential O exists, in the forms du dt' vc + wri= — iA- *> T ' cx Wt -wZ+uC= dw . dV •(5) provided H dp +^ 2 + Q, .(6) * See Nanson, Mess, of Math. hi. 120 (1874); Kirchhoff, Mechanik, c. xv. (1876) Papers, ii. 47 (1883). t Nanson, I.e. Stokes, 206 Vortex Motion [chap, vii where q 2 = i£ l + v 2 + w 2 . From the second and third of these we obtain, eliminating % by cross-differentiation, Remembering the relation ^ + J- + ^=0, (7) and the equation of continuity Dp (du dv dw\ . . we easily deduce the first of equations (4). To interpret these equations we take, at time t, a linear element whose projections on the co-ordinate axes are to, ty, &»-*£, c2, c^, (9) P P P where e is infinitesimal. If this element be supposed to move with the fluid, the rate at which dx is increasing is equal to the difference of the values of u at the two ends, whence D8x £du ndu tdu Dt pdx pdy p oz It follows, by (4), that *(*-JH »(*-JH *(H)-° (10) Helmholtz concludes that if the relations (9) hold at time £, they will hold at time t + dt, and so on, continually. The inference is, however, not quite rigorous; it is in fact open to the criticisms which Stokes* directed against various defective proofs of Lagrange's velocity-potential theorem f. By way of establishing a connection with Kelvin's investigation we may notice that the equations (2) express that the circulation is constant in each of three infinitely small circuits initially perpendicular, respectively, to the three co-ordinate axes. Taking, for example, the circuit which initially bounded the rectangle 8b 8c, and denoting by A, B, C the areas of its projections at time t on the co-ordinate planes, we have 4-|M»te, B=l%f>SbSc, C- d J*£»ic d (6, c) d (6, c) d (6, c) so that the first of the equations referred to is equivalent + to §A+ n B+£C=£ 8b8c (11) As an application of the equations (4) we may consider the motion of a liquid of uniform vorticity contained in a fixed ellipsoidal vessel §. The formulae u=qz — ry, v = rx—pz, w—py — qx (12) * I.e. ante p. 17. f It may be mentioned that, in the case of an incompressible fluid, equations somewhat similar to (4) had been established by Lagrange, Miscell. Taur. ii. (1760) [Oeuvres, i. 442]. The author is indebted for this reference, and for the above remark on Helmholtz' investigation, to Sir J. Larmor. Equations equivalent to those given by Lagrange were obtained independently by Stokes, I.e., and made the basis of a rigorous proof of the velocity-potential theorem. J Nanson, Mess, of Math. vii. 182 (1878). A similar interpretation of Helmholtz' equations was given by the author of this work in the Mess, of Math. vii. 41 (1877). Finally it may be noted that another proof of Lagrange's theorem, based on elementary dynamical principles, without special reference to the hydrokinetic equations, was indicated by Stokes, Camb. Trans, viii. [Papers, i. 113], and carried out by Kelvin in his paper on Vortex Motion. § Cf. Voigt, "Beitrage zur Hydrodynamik," G'dtt. Nachr. 1891, p. 71; Tedone, Nuovo Cimento, xxxiii. (1893). The artifice in the text is taken from Poincare, "Sur la precession des corps deformables," Bull. Astr. 1910. 146-147] Helmholtz' Equations 207 obviously represent a uniform rotation of the fluid as a solid within a spherical boundary. Transforming the co-ordinates and the corresponding velocities by homogeneous strain we obtain the formulae u qz ry v _rx pz w _py qx . . a cbba ceo a as representing a certain motion within a fixed ellipsoidal boundary £+£+£-1 (") 2 ' ft* ' c Thesemake «=g + £>, ,-£ + *)* f-g + |)r (15) Substituting in (4) we obtain (P+^ff-P 1 -*)^, (16) which may be written a 2 (6 2 +c 2 )^={6 2 (c 2 + a 2 )-c 2 (a 2 -|-6 2 )}^, (17) with two similar equations. We have here an identity as to form with Euler's equations of free motion of a solid about a fixed point. We easily deduce the integrals f 2 +p + §= const -> (18) 6V£ 2 c 2 «V a 2 ^ 2 ^ , , im and £TT^+ 2 , 2 + o , ,, =const., (19) the former of which is a verification of one of Helmholtz' theorems, whilst the latter follows from the constancy of the energy. 147. It is easily seen by the same kind of argument as in Art. 41 that no continuous irrotational motion is possible in an incompressible fluid filling infinite space, and subject to the condition that the velocity vanishes at infinity. This leads at once to the following theorem : The motion of a fluid which fills infinite space, and is at rest at infinity, is determinate when we know the values of the expansion (0, say) and of the component vorticities f , r) y f, at all points of the region. For, if possible, let there be two sets of values, u 1} v 1} w 1} and u 2 , tfo, w%, of the component velocities, each satisfying the equations du dv dw n T* + r y + is =0 > w dw dv _ £ du dw _ dv du _ y . dy~dz~~*' di~d^~ V} dx~dy~^ {> throughout infinite space, and vanishing at infinity. The quantities u' = u 1 — u 2 , v' = vx — v z , w , = w 1 — w 2 will satisfy (1) and (2) with 6, f, 77, ? = 0, and will- vanish at infinity. Hence, in virtue of the result above stated, they will everywhere vanish, and there is only one possible motion satisfying the given conditions. In the same way we can shew that the motion of a fluid occupying any limited simply-connected region is determinate when we know the values of 208 Vortex Motion [chap, vii the expansion, and of the component vorticities, at every point of the region, and the value of the normal velocity at every point of the boundary. In the case of an w-ply-connected region we must add to the above data the values of the circulations in n several independent circuits of the region. 148. If, in the case of infinite space, the quantities 6, f, r), f all vanish beyond some finite distance of the origin, the complete determination of u, v, w in terms of them can be effected as follows *. The component velocities due to the expansion can be written down at once from Art. 56 (1), it being evident that the expansion 6' in an element 8x'8y'8z' is equivalent to a simple source of strength O'Bx'Sy'Sz'. We thus obtain 83> d® d<$> u = ~Tx> v = -fy> W = -Tz> (1) where <S> =~ [[[-dx'dy'dz', (2) r denoting the distance between the point (V, y\ z') at which the volume- element of the integral is situate and the point (x, y, z) at which the values of u, v, w are required, viz. r = {(x - x'f + (y- y'f + (z - z'f)$. The integration includes all parts of space at which 6 r differs from zero. To find the velocities due to the vortices, we note that when there is no expansion, the flux across any two open surfaces bounded by the same curve as edge will be the same, and will therefore be determined solely by the configuration of the edge. This suggests that the flux through any closed curve may be expressed as a line-integral taken round the curve, say J(Fdx+Gdy + Hdz) (3) On this hypothesis we should have, by the method of Art. 31, u=z dH_dG v = dF_dH w = ^_^l / 4) ~" dy dz ' dz dx ' dx dy It is necessary and (as we have seen) sufficient that the functions F, G, H should satisfy djv_dv = l/dF + dG + dH\ V2F dy dz dx \ dx dy dz J together with two similar equations. They will in any case be indeterminate to the extent of three additive functions of the forms d%\dx, dx/dy, dx/dz, respectively, and we may, if we please, suppose x to be chosen so that dF dG dH ,-x ^-- + ^- + ^- = 0, (5) dx dy dz * The investigation which follows is substantially that given by Helmholtz. The kinematical problem in question was first solved, in a slightly different manner, by Stokes, " On the Dynamical Theory of Diffraction," Camb. Trans, ix. (1849) [Papers, ii. 254...]. 47r J J J r 147-148] Velocities due to a Vortex-System 209 in which case V 2 ^=-?, V 2 G = - V , ^ 2 H=-^ (6) Particular solutions of these equations are obtained by equating F, G, H to the potentials of distributions of matter whose volume-densities are f/47r, 77/473-, ?/47r, respectively; thus ? dx'dy'dz', G = ~ [ I [ £ dx'dy'dz', H = ^ [ [ j £ dx'dy'dz, (7) where the accents attached to f, 77, f are used to distinguish the values of these quantities at the point {x ', y', 2'). The integrations are to include, of course, all places where f, 97, f differ from zero. It remains to shew that these values of F, G, H do in fact satisfy (5). Since d/cx . r _1 = — d/dx'. r~\ the formulae (7) make dF dG dH 1 (ff/ H 3 1 ,81 w ai\, u ,,, The right-hand member vanishes, by a generalization of the theorem of Art. 42 (4)*, since 3d? 3y 80 everywhere, whilst Zf -f 77177 + ?if = at the surfaces of the vortices (where f, 77, J may be discontinuous), and f , 77, f vanish at infinity. The complete solution of our problem is obtained by superposition of the results contained in (1) and (4), viz. 3# 9y 3s ?J = _^ + ^_^ 3y dz d% ' = _3<£ s^_M! 82 3a? 3y where <£, F, G, # have the values given in (2) and (7). It may be added that the proviso that 0, f, 77, f should vanish beyond a certain distance from the origin is not absolutely essential. It is sufficient if the data be such that the integrals in (2) and (7), when taken over infinite space, are convergent. This will certainly be the case if 6, f, 77, £ are ultimately of the order R~ n , where R denotes distance from the origin, and n >3f. When the region occupied by the fluid is not unlimited, but is bounded (in whole or in part) by surfaces at which the normal velocity is given, and when further (in the case of an n-ip\y connected region) the value of the circulation in each of n independent circuits is prescribed, the problem may The singularity which occurs at the point r = is assumed to be treated here and elsewhere as in the theory of Attractions. The result is not affected, t Cf. Leathern, Cambridge Tracts, No. 1 (2nd ed.), p. 44. 210 Vortex Motion [chap, vii by a similar analysis be reduced to one of irrotational motion, of the kind considered in Chapter III., and there proved to be determinate. This may be left to the reader, with the remark that if the vortices traverse the region, beginning and ending on the boundary, it is convenient to imagine them continued beyond it, or along the boundary, in such a manner that they form re-entrant filaments, and to make the integrals (7) refer to the complete system of vortices thus obtained. On this understanding the condition (5) will still be satisfied. There is an exact correspondence between the analytical relations above developed and certain formulae in Electro-magnetism. If, in the equations (1) and (2) of Art. 147, we write a, 3, y, p, u, % w, p for u, v t w, 0, |, Tj, C, 0, respectively, we obtain da 9/3 dy dx^dy^dz~ 9 ' dy w 93 9o dy _ 83 dz dx ' dx .(9) da which are the fundamental relations of the theory referred to ; viz. o, 3, y are the compo- nents of magnetic force, u, v, w those of electric current, and p is the volume-density of the imaginary magnetic matter by which any magnetization present in the field may be repre- sented* Hence, the vortex- filaments correspond to electric circuits, the strengths of the vortices to the strengths of the currents in these circuits, sources and sinks to positive and negative magnetic poles, and, finally, fluid velocity to magnetic force t. The analogy will of course extend to all results deduced from the fundamental relations ; thus, in equations (8), <J> corresponds to the magnetic potential and F, G, H to the com- ponents of 'electro-magnetic momentum.' 149. To interpret the result contained in Art. 148 (8), we may calculate the values of u, v, w due to an isolated re-entrant vortex-filament situate in an infinite mass of incompressible fluid which is at rest at infinity. Since = 0, we shall have <E> = 0. Again, to calculate the values of F, G, H, we may replace the volume-element Sx'Sy'Bz' by cr'Ss', where Ss r is an element of the length of the filament, and a its cross-section. Also , ,dx , r dy' , ,dz' Z=°>d?> v=03 d7> ?=G, 57' where a>' is the vorticity. Hence the formulae (7) of Art. 148 become F = —[— Q = JL\d]L h=—[— (1) 47rJ r ' 47rJ r ' 47rJ r ' where k, = ©V, measures the strength of the vortex, and the integrals are to be taken along the whole length of the filament. * Cf . Maxwell, Electricity and Magnetism, Art. 607. The analogy has been improved by the adoption of the 'rational' system of electrical units advocated by Heaviside, Electrical Papers, London, 1892, i. 199. t This analogy was first pointed out by Helmholtz ; it has been extensively utilized by Kelvin in his papers on Electrostatics and Magnetism. 148-150] Velocities due to an isolated Vortex 211 Hence, by Art. 148 (4), we have with similar results for v, w. We thus find* tc [(dy z — z dz r y — y'\ ds' u = 47r I \ds' r ds' ?■ dz' x — x' dx' z — z'\ ds' ds' r ds' r ) r 2 ^k^^-^^i^.v (2) k Udx dx' y — y' dy' x — x'\ ds' w- , , , 2 r ds r J it If 8u, 8v, 8w denote the parts of these expressions which involve the element 8s' of the filament, it appears that the resultant of 8u, Sv, 8w is perpendicular to the plane containing the direction of the vortex-line at (x, y', z') and the line r, and that its sense is that in which the point (x, y, z) would be carried if it were attached to a rigid body rotating with the fluid element at (x\ y', z'). For the magnitude of the resultant we have {(3^ + (8^ + (^|i=^ sin ^' ) (3) where x is tne angle which r makes with the vortex-line at (V, y', z'). With the change of symbols indicated in the preceding Art. this result becomes identical with the law of action of an electric current on a magnetic pole +. Velocity -Potential due to a Vortex. 150. At points external to the vortices there exists a velocity-potential, whose value may be obtained as follows. Taking for shortness the case of a single re-entrant vortex, we have, from the preceding Art., in the case of an incompressible fluid, "'iiWr-V-bW (1) By Stokes' Theorem (Art. 32 (4)) we can replace a line-integral extending round a closed curve by a surface-integral taken over any surface bounded by that curve; viz. we have, with a slight change of notation, j W+w+ ^,=j|{«(f-g) + .(g-g) + .(i-i)}-- If we put P = 0, Q = S-,~, R = -^-,- oz r : "-' we find dJl_°Q = d 2 il_ dP dR = d 2 1 dy' dz' dx'*r" dz' dx ~ dx'dy' r' ' * These are equivalent to the forms obtained by Stokes, I.e. ante p. 208. f Ampere, Theorie mathematique des phenomenes electro-dynamiques, Paris, 1826. d dy' 1 dQ dx' dP a 2 dx'dz' 1 212 Vortex Motion [chap, vii so that (1) may be written U = £r\\{ , 3 3 3 \ d 1 7af Hence, and by similar reasoning, we have, since d/daf. r~ x = — d/dx . r -1 , — 1> — |. —I < 2 > where * = £ JJ(^ + ™|> + «^) ^ (3) Here I, m, w denote the direction-cosines of the normal to the element 8S' of a surface bounded by the vortex-filament. The formula (3) may be otherwise written cos^ *=£// dsr, (4) where S- denotes the angle between r and the normal (I, m, n). Since cos ^dS'/r 2 measures the elementary solid angle subtended by 8S' at (x, y, z), we see that the velocity-potential at any point, due to a single re-entrant vortex, is equal to the product of k/^tt into the solid angle which a surface bounded by the vortex subtends at that point. Since this solid angle changes by 4>tt when the point in question describes a circuit embracing the vortex, we verify that the value of <f> given by (4) is cyclic, the cyclic constant being tc. Cf. Art. 145. It may be noticed that the expression in (4) is equal to the flux (in the negative direction) through the aperture of the vortex, due to a point-source of strength k at the point (a?, y, z). Comparing (4) with Art. 56 (4) we see that a vortex is, in a sense, equivalent to a uniform distribution of double sources over any surface bounded by it. The axes of the double sources must be supposed to be everywhere normal to the surface, and the density of the distribution to be equal to the strength of the vortex. It is here assumed that the relation between the positive direction of the normal and the positive direction of the axis of the vortex-filament is of the 'right-handed' type. See Art. 31. Conversely, it may be shewn that any distribution of double sources over a closed surface, the axes being directed along the normals, may be replaced by a system of closed vortex-filaments lying in the surface*. The same thing will appear independently from the investigation of the next Art. Vortex-Sheets. 151. We have so far assumed u, v, w to be continuous. We may now shew how cases where surfaces of discontinuity present themselves may be brought within the scope of our theorems. * Cf. Maxwell, Electricity and Magnetism, Arts. 485, 652. i50-i5i] Vortex-Sheets 213 The case of a discontinuity in the normal velocity alone has already been treated in Art. 58. If u, v, w denote the component velocities on one side, and u\ v\ w' those on the other, it was found that the circumstances could be represented by imagining a distribution of simple sources, with surface- density l{u'—u) + m (v f — v) + n {w' — w), where I, ra, n denote the direction-cosines of the normal drawn towards the side to which the accents refer. Let us next consider the case where the tangential velocity (only) is dis- continuous, so that l(u'-u) + m(v'-v)+n(w'-<w) = (1) We will suppose that the lines of relative motion, which are defined by the differential equations dx dy dz (i> are traced on the surface, and that the system of orthogonal trajectories to these lines is also drawn. Let PQ, P'Q' be linear elements drawn close to the surface, on the two sides, parallel to a line of the system (2), and let PP' and QQ' be normal to the surface and infinitely small in comparison with PQ or P'Q'. The circulation in the circuit P'Q'QP will then be equal to (q'—q) PQ, where q, q' denote the absolute velocities on the two sides. This is the same as if the position of the surface were occupied by an infinitely thin stratum of vortices, the orthogonal trajectories above-mentioned being the vortex- lines, and the vorticity co and the (variable) thickness 8n of the stratum being connected by the relation (o8n = q' — q (3) The same result follows from a consideration of the discontinuities which occur in the values of u, v, w as determined by the formulae (4) and (7) of Art. 148, when we apply these to the case of a stratum of thickness 8n within which £, t), f are infinite, but so that £8n, rjBn, ^8n are finite*. It was shewn in Arts. 147, 148 that any continuous motion of a fluid filling infinite space, and at rest at infinity, may be regarded as due to a suitable arrangement of sources and vortices distributed with finite density. We have now seen how by considerations of continuity we can pass to the case where the sources and vortices are distributed with infinite volume- density, but infinite surface-density, over surfaces. In particular, we may take the case where the infinite fluid in question is incompressible, and is divided into two portions by a closed surface over which the normal velocity is con- tinuous, but the tangential velocity discontinuous, as in Art. 58 (12). This is * Helmholtz, I.e. ante p. 202. 214 Vortex Motion [chap, vii equivalent to a vortex-sheet; and we infer that every continuous irrotational motion, whether cyclic or not, of an incompressible substance occupying any region whatever, may be regarded as due to a certain distribution of vortices over the boundaries which separate it from the rest of infinite space. In the case of a region extending to infinity, the distribution is confined to the finite portion of the boundary, provided the fluid be at rest at infinity. This theorem is complementary to the results obtained in Art. 58. The foregoing conclusions may be illustrated by means of the results of Art. 91. Thus when a normal velocity S n was prescribed over the sphere r=a, the values of the velocity- potential for the internal and external space were found to be ♦-;©"*. -♦~ 5 j I gT*. respectively. Hence if 8e be the angle which a linear element drawn on the surface subtends at the centre, the relative velocity estimated in the direction of this element will be 2n + l &S; n(n + l) de ' The resultant relative velocity is therefore tangential to the surface, and perpendicular to the contour lines (S n = const.) of the surface-harmonic S n , which are therefore the vortex- lines. For example, if we have a thin spherical shell filled with and surrounded by liquid, moving as in Art. 92 parallel to the axis of x, the motion of the fluid, whether internal or external, will be that due to a system of vortices arranged in parallel circles on the sphere ; the strength of an elementary vortex being proportional to the projection, on the axis of %, of the breadth of the corresponding zone of the surface*. Impulse and Energy of a Vortex-System. 152. The following investigations relate to the case of a vortex-system of finite dimensions in an incompressible fluid which fills infinite space and is at rest at infinity. The problem of finding a distribution of impulsive force (X', Y', Z') per unit mass which would generate the actual motion (u, v, w) instantaneously from rest is to some extent indeterminate, but a sufficient solution for our purpose may be obtained as follows. We imagine a simply-connected surface S to be drawn enclosing all the vortices. We denote by </> the single-valued velocity-potential which obtains outside S, and by <j> x that solution of V 2 <£ = which is finite throughout the interior of S, and is continuous with <f> at this surface. In other words, </>i is the velocity -potential of the motion which would be produced within $ by the application of impulsive pressures p<f> over the surface. If we now assume *-+£. *--+£• z '=™ + t « * The same statements hold also for an ellipsoidal shell moving parallel to one of its principal axes. See Art. 114. 151-152] Impulse of a Vortex-System 215 at internal points, and Z'=0, 7'=0, Z'=0 (2) at external points, it is evident on reference to Art. 11 that these forces would in fact generate the actual motion instantaneously from rest, the distribution of impulsive pressure being given by p<f> at externa], and pfa at internal, points. The forces are discontinuous at the surface, but the discontinuity is only in the normal component, the tangential components vanishing just inside and just outside owing to the continuity of </> with fa. Hence if (I, m, n) be the direction-cosines of the inward normal, we should have rf-nF'=0, nX'-lZ'=0, ZF'-mX'=0, (3) at points just inside the surface. Now if we integrate over the volume enclosed by S we have jjj(yr- •») dxdydz=\\\\y g - g) -. g - g)} dxdydz = -fJ{y(lY , -mX , )-z(nX , -lZ , )}dS + 2jfJX , dxdydz, (4) where the surface-integral vanishes in virtue of (3). Again - jj}(2/ 2 + * 2 ) f dxdydz = - JJJ(y»+ *■) (g' - 1) d«4y<b = tf(y*+z 2 )(mZ'--nY')dS + 2fff{yZ'-zY')dxdydz, (5) where the surface-integral vanishes as before. We thus obtain for the force- and couple-resultants of the impulse of the vortex-system the expressions •P = ipfff(yS ~ z v) dxdydz, L = -\pjjj(y 2 + z 2 ) ^dxdydz; Q=ipJJK*Z-*S) dxdydz, M=-yfff(z*+x*) v dxdydz,r...(6) R = ipIII( x V - y%) dxdydz, N = -ipffj(x 2 + y 2 ) {dxdydz. To apply these to the case of a single re-entrant vortex-filament of infinitely small section a, we replace the volume element by a 8s, and write j. dx dy ., dz ,,_. i = ( °ds> V = ( °ts> ^ =( °ds (0 Hence P = ipco<rf(ydz -zdy) = Kpffl'dS', (8) L = -ycoaj(y 2 + z 2 )dx = -fcpjf(m , z-n , y)dS', (9) with similar formulae. The line-integrals are supposed to be taken along the filament, and the surface-integrals over a barrier bounded by it, and V , m', n are the direction-cosines of the normal to an element 8S' of the barrier. The Z-%-W + F(t) (2) 216 Vortex Motion [chap, vii identities of the different forms follow from Stokes' Theorem. We have also written k for cocr, i.e. tc is the circulation round the filament*. The whole investigation has reference of course to the instantaneous state of the system, but it may be recalled that, when no extraneous forces act, the impulse is, by the argument of Art. 119, constant in every respect. 153. Let us next consider the energy of the vortex-system. It is easily proved that under the circumstances presupposed, and in the absence of extraneous forces, this energy will be constant. For if T be the energy of the fluid bounded by any closed surface S, we have, putting V=0 in Art. 10 (5), DT -jr- = Jf(lu + mv + nw) pdS (1) If the surface S enclose all the vortices, we may put p _ d(j> P and it easily follows from Art. 150 (4) that at a great distance R from the vortices p will be finite, and lu + mv -f nw of the order R~ z , whilst when the surface S is taken wholly at infinity, the elements 8S vary as R 2 . Hence, ultimately, the right-hand side of (1) vanishes, and we have T= const (3) We proceed to investigate one or two important kinematical expressions for T, still confining ourselves, for simplicity, to the case where the fluid (sup- posed incompressible) extends to infinity, and is at rest there, all the vortices being within a finite distance of the origin. The first of these expressions is indicated by the electro-magnetic analogy pointed out in Art. 148. Since = 0, and therefore <I> = 0, we have 2T=pfff(u 2 + v 2 + w 2 ) dxdydz = p \\\{ u (f - S) + v (£- s) + w (£- !)} dxdydz > by Art. 148 (4). The last member may be replaced by the sum of a surface- integral pff{F(mw — nv)+ G (nu — Iw) + H{lv- mu)} dS, and a volume-integral * The expressions (8) and (9) were obtained by elementary reasoning by J. J. Thomson, On the Motion of Vortex Rings (Adams Prize Essay), London, 1883, pp. 5, 6, and the formulae (6) deduced from them, with, however, the opposite signs in the case of L, M, N. The correction is due to Mr Welsh. An interesting test of the formulae as they now stand is afforded by the case of a spherical mass rotating as if solid and surrounded by fluid at rest, provided we take into account the spherical vortex-sheet which represents the discontinuity of velocity. 152-153] Energy of a Vortex- System 217 At points of the infinitely distant boundary, F, G, H are ultimately of the order R~ 2 . and u, v, w of the order R~ z , so that the surface-integral vanishes, and we have T=i P JfJ(FS+Q v + H0dxdydz (4) or, substituting the values of F, 0, H from Art. 148 (7), T " t JJJjJJ ~ "r' + rr *"****'<&*' (*) where each volume- integration extends over the whole space occupied by the vortices. A slightly different form may be given to this expression as follows. Regarding the vortex-system as made up of filaments, let 8s, 8s' be elements of length of any two filaments, cr, a the corresponding cross-sections, and co, co' the corresponding vorticities. The elements of volume may be taken to be cr8s and cr'8s\ respectively, so that the expression following the integral signs in (5) is equivalent to C0S€ a„ ' '*«' . coats . co <t 6s , r where e is the angle between 8s and 8s' . If we put co<t — k, co'a' — k , we have T " ir XKK ' \\ ^ dsdS ' (6) where the double integral is to be taken along the axes of the filaments, and the summation 2 includes (once only) every pair of filaments which are present. The factor of p in (6) is identical with the expression for the energy of a system of electric currents flowing along conductors coincident in position with the vortex-filaments, with strengths k, k', ... respectively*. The above investigation is in fact merely an inversion of the argument given in treatises on Electro-magnetism, whereby it is proved that ^2ii' f f^ dsds' = h f f f(a*+p 2 + y 2 )dxdydz, where i, % denote the strengths of the currents in the linear conductors whose elements are denoted by 8s, 8s' , and a, 3, y are the components of magnetic force at any point of the field. The theorem of this Art. is purely kinematical, and rests solely on the assumption that the functions u, v, w satisfy the equation of continuity, du dv cw _ Ci dx dy dz ' throughout infinite space, and vanish at infinity. It can therefore by an easy generaliza- tion be extended to a case considered in Art. 144, where a liquid is supposed to circulate irrotationally through apertures in fixed solids, the values of u, v, w being now taken to be zero at all points of space not occupied by the fluid. The investigation of Art. 151 shews that the distribution of velocity thus obtained may be regarded as due to a system of vortex-sheets coincident with the bounding surfaces. The energy of this system will be given by an obvious adaptation of the formula (6) above, and will therefore be proportional to that of the corresponding system of electric current-sheets. This proves a statement made by anticipation in Art. 144. * The ' rational ' system of electrical units being understood ; see ante p. 210. 218 Vortex Motion [chap, vii Under the circumstances stated at the beginning of Art. 152, we have another useful expression for T\ viz. T— pfff{u(y£— zrj) + v (zg — x£) -hw(xr) — y%)} dxdydz* (7) To verify this, we take the right-hand member, and transform it by the process already so often employed, omitting the surface-integrals for the same reason as in the preceding Art. The first of the three terms gives p S\S u [ y (£ - D - * £ - £)} dxdydz ~ ~~ P ) ( v y "** wz } a — u \ d°°dydz. dx j Transforming the remaining terms in the same way, adding, and making use of the equation of continuity, we obtain p I ( w 2 ■+- v 2 + w 2 + xu x- + yv =- + zw ~- J dxdydz, or, finally, on again transforming the last three terms, ipfff(u 2 + v 2 + w 2 ) dx dydz. In the case of a finite region the surface-integrals must be retained"]-. This involves the addition to the right-hand side of (7) of the term pfS {(^ u + mv + nw) {xu + yv + zw) — J (Ix 4- my + nz) q 2 \ dS, (8) where q 2 = u 2 + v 2 + w 2 . This simplifies in the case of a fixed boundary. The value of the expression (7) must be unaltered by any displacement of the origin of co-ordinates. Hence we must have ///(«?— wrf) dx dy dz — 0, ///( w% — u%) dx dy dz = 0, JJf(u7j — v%) dx dy dz = 0. (9) These equations, which may easily be verified by partial integration, follow also from the consideration that if there are no extraneous forces the components of the impulse parallel to the co-ordinate axes must be constant. Thus, taking first the case of a fluid enclosed in a fixed envelope of finite size, we have, in the notation of Art. 152, P=pjjfudxdydz-pjfl<pdS, (10) if cp denote the velocity-potential near the envelope, where the motion is irrotational. Hence d ± =p j j J " *? dxdydz- P j fl^dS = ~ p l I \% dxd y dz + p\ I {(vC-wt)) dxdydz- pi (l^dS, (11) by Art. 146 (5). The first and third terms of this cancel, since at the envelope we have x' = dcp/dt, by Art. 20 (4) and Art. 1 46 (6). Hence for any re-entrant system of vortices enclosed in a fixed vessel, we have dP -^=pjjj(v£-wri) dxdydz, (12) witb two similar equations. It has been proved in Art. 119 that if the containing vessel be infinitely large, and infinitely distant from the vortices, P is constant. This gives the first of equations (9). * Motion of Fluids, Art. 136 (1879). t J. J. Thomson, I.e. ante p. 216. 153-154] Rectilinear Vortices 219 Conversely from (9), established otherwise, we could infer the constancy of the com- ponents P, Q, R of the impulse*. Rectilinear Vortices. 154. When the motion is in two dimensions x, y we have w = 0, whilst u, v are functions of x, y, only. Hence f = 0, rj = 0, so that the vortex-lines are straight lines parallel to *. The theory then takes a very simple form. The formulae (8) of Art. 148 are now replaced by dx dy ' dy dx ' the functions <f>, yfr being subject to the equations V^ — 0, W-C (2) where V ^ 2 = ^ + ^' and to the proper boundary-conditions. In the case of an incompressible fluid, to which we will now confine our- selves, we have >' U = -fy> V = Tx> (3) where ty is the stream-function of Art. 59. It is known from the Theory of Attractions that the solution of V 1 2 ^ = ?, (4) X being a given function of x y y, is * = ^//?'logr<fe'A/ + *o. (5) where f denotes the value of f at the point (x' } y'), and r stands for {( x - x >f + (y-y'f}l. The 'complementary function' ^ may he any solution of V! 2 ^ = 0; (6) it enables us to satisfy the boundary-conditions. In the case of an unlimited mass of liquid, at rest at infinity, -^ is constant. The formulae (3) and (5) then give »=-^[[?'^W, **ki\r m ^*** < 7) Hence a vortex-filament whose co-ordinates are x', y' and whose strength is k contributes to the motion at (x, y) a velocity whose components are k v — y' , tc x — x' 2tt r 2 27r r 2 This velocity is perpendicular to the line joining the points (x, y), (x' } y'), and its amount is KJIirr. * J. J. Thomson, I.e. 220 Vortex Motion [chap, vii Let us calculate the integrals ffu^dxdy, and ffvgdxdy, where the integra- tions include all portions of the plane xy for which f does not vanish. We have u Zdcody = ~ jjjfe' V -^/ dxdydx'dy', where each double integration includes the sections of all the vortices. Now, corresponding to any term &' y -^fdxdydx'dy' of this result, we have another KK' 2 dxdydx'dy', and these two neutralize each other. Hence, and by similar reasoning, ffu£dxdy = 0, Jfv£dxdy = (8) If as before we denote the strength of a vortex by k, these results may be written 2fcu = 0, Xkv = (9) Since the strength of each vortex is constant with regard to the time, the equations (9) express that the point whose co-ordinates are -S- *-g ™ is fixed throughout the motion. This point, which coincides with the centre of inertia of a film of matter distributed over the plane xy with the surface-density f, may be called the 'centre' of the system of vortices, and the straight line parallel to z of which it is the projection may be called the 'axis' of the system. If Xk — } the centre is at infinity, or else indeterminate. 155. Some interesting examples are furnished by the case of one or more isolated vortices of infinitely small section. Thus : 1°. Let us suppose that we have only one vortex-filament present, and that the vorticity f has the same sign throughout its infinitely small section. Its centre, as just defined, will lie either within the substance of the filament, or infinitely close to it. Since this centre remains at rest, the filament as a whole will be stationary, though its parts may experience relative motions, and its centre will not necessarily lie always in the same element of fluid. Any particle at a finite distance r from the centre of the filament will describe a circle about the latter as axis, wftth constant velocity kJ^ttv. The region external to the vortex is doubly-connected; and the circulation in any (simple) circuit embracing it is of course k. The irrotational motion of the surrounding fluid is the same as in Art. 27 (2). ]54 -155] Vortex-Pair 221 2°. Next suppose that we have two vortices, of strengths k\ } tc 2 , respectively. Let A , B be their centres, the centre of the system. The motion of each filament as a whole is entirely due to the other, and is therefore always per- pendicular to AB. Hence the two filaments remain always at the same distance from one another, aod rotate with constant angular velocity about 0, which is fixed. This angular velocity is easily found; we have only to divide the velocity of A (say), viz. # a /(27r. AB), by the distance AO, where /e 2 AO /C1 + /C2 AB, and so obtain Lit . A. if If «i, k 2 be of the same sign, i.e. if the directions of rotation in the two vortices be the same, lies between A and B; but if the rotations be of opposite signs, lies in AB, or B A, produced. If k±= ■ — k 2 , is at infinity; but it is easily seen that A, B move with equal velocities k^(2it . AB) at right angles to AB, which remains fixed in direction. Such a combination of two equal and opposite vortices may be called a 'vortex-pair.' It is the two-dimensional analogue of a circular vortex- ring (Art. 160), and exhibits many of the properties of the latter. The stream-lines of a vortex-pair form a system of coaxal circles, as shewn on p. 67, the vortices being at the limiting points (+ a, 0). To find the relative stream-lines, we superpose a general velocity equal and opposite to that of the vortices, and obtain, for the relative stream-function, r 2tt V2a T '<;)■ .(i) in the notation of Art. 64, 2°. The figure (which is turned through 90° for convenience) shews a few of the lines. The line yfr — consists partly of the axis of y, and partly of an oval surrounding both vortices. 222 Vortex Motion [chap, vii It is plain that the particular portion of fluid enclosed within this oval accompanies the vortex-pair in its career, the motion at external points being exactly that which would be produced by a rigid cylinder having the same boundary; cf. Art. 71. The semi-axes of the oval are 2'09 a and 1*73 a, approximately *. A difficulty is sometimes felt, in this as in the analogous instance of a vortex-ring, in understanding why the vortices should not be stationary. If in the figure on p. 70 the filaments were replaced by solid cylinders of small circular section, the latter might indeed remain at rest, provided they were rigidly connected by some contrivance which did not interfere with the motion of the fluid ; but in the absence of such a connection they would in the first instance be attracted towards one another, on the principle explained in Art. 23. This attraction is however neutralized if we superpose a general velocity V of suitable amount in the direction opposite to the cyclic motion half-way between the cylinders. To find V, we remark that the fluid velocities at the two points (a±c, 0), where c is small, will be approximately equal in absolute magnitude, provided V+— — = — +— - V 2irC 4-rra 2irc Ana ' where k is the circulation. Hence V Ana which is exactly the velocity of translation of the vortex-pair, in the original form of the problem t. Since the velocity of the fluid at all points of the plane of symmetry is wholly tangential, we may suppose this plane to form a rigid boundary of the fluid on either side of it, and so obtain the case of a single rectilinear vortex in the neighbourhood of a fixed plane wall to which it is parallel. The filament moves parallel to the plane with the velocity KJ^rrh, where h is the distance from the wall. Again, since the stream-lines are circles, we can also derive the solution of the case where we have a single vortex-filament in a space bounded, either internally or externally, by a fixed circular cylinder. Thus, in the figure, let EPD be the section of the cylinder, A the position of the vortex (supposed in this case external), and let B be the 'image' of A with respect to the circle EPD, viz. C being the centre, let CB.CA=c 2 , where c is the radius of the circle. If P be any point on the circle, we have AP _AE _ AD_ BP~ BE~ BD~° ' so that the circle occupies the position of a stream-line due to a vortex-pair at A, B. Since the motion of the vortex A would be perpendicular to AB, * Cf. Sir W. Thomson, "On Vortex Atoms," Phil. Mag. (4), xxxiv. 20 (1867) [Papers, iv. 1]; and Eiecke, Gott. Nachr. 1888, where paths of fluid particles are also delineated. t A more exact investigation is given by Hicks, "On the Condition of Steady Motion of Two Cylinders in a Fluid," Quart. Journ. Math. xvii. 194 (1881). 155] Special Cases 223 it is plain that all the conditions of the problem will be satisfied if we suppose A to describe a circle about the axis of the cylinder with the constant velocity < k.CA 2n.AB~ $tt(CA 2 -c 2 )' where k denotes the strength of A. In the same way a single vortex of strength k, situated inside a fixed circular cylinder, say at B, would describe a circle with constant velocity k.CB 2tt(c 2 -CB 2 )' It is to be noticed, however* that in the case of the external vortex the motion is not completely determinate unless, in addition to the strength k, the value of the circulation in a circuit embracing the cylinder (but not the vortex) is prescribed. In the above solution, this circulation is that due to the vortex-image at B and is — k. This may be annulled by the superposition of an additional vortex + k at (7, in which case we have, for the velocity of A, K.CA K KC 2 ~ 2rr(CA 2 ~c 2 ) + 27r.CA 2tt . CA (CA 2 -C 2 ) ' For a prescribed circulation k we must add to this the term k'\2tt . CA. L. Foppl t, using the method of images, has investigated the case of a cylinder advancing through fluid with velocity U, and followed by a vortex-pair symmetrically situated with respect to the line of advance of the centre. It appears that the vortices can maintain their position relative to the cylinder provided they lie on the curve 2ry = r 2 — a 2 , and that the strengths of the vortices corresponding to a given position on this curve are + 2 *0-3) He finds, however, that the arrangement is unstable for anti-symmetrical disturbances. Some paths of vortices in a stream past a cylindrical obstacle (with circulation) have been traced by Walton J. The path of a vortex in a semicircular region is investigated by K. De § by Routh's method referred to on p. 224. 3°. If we have four parallel rectilinear vortices whose centres form a rectangle ABB' A', the strengths being k for the vortices A' } B, and — k for the vortices A, B\ it is evident that the centres will always form a rectangle * F. A. Tarleton, "On a Problem in Vortex Motion," Proc. R. I. A. December 12, 1892. f " Wirbelbewegung hinter einem Kreiszylinder," Sitzb. d. k. bdyr. Akad. d. Wiss. 1913. | Proc. R. I. Acad, xxxviii. A (1928). § Bull, of the Calcutta Math. Soc. xxi. 197 (1929). 224 Vortex Motion [chap, vii Further, the various rotations having the directions indicated in the figure, we see that the effect of the presence of the pair A, A' on B, B' is to separate them, and at the same time to diminish their velocity perpendicular to the line joining them. The planes which bisect AB, AA' at right angles may (either or both) be taken as fixed rigid boundaries. We thus get the case where a pair of vortices, of equal and opposite strengths, move towards (or from) a plane wall, or where a single vortex moves in the angle between two perpendicular walls. If x, y be the co-ordinates of the vortex A relative to the planes of symmetry, we readily find . <_ #2 -_ji_ y^_ /on X ~ 4n-yr*> V ~^' xr 2 " { } where r 2 — x 2 -\~y 2 . By division we obtain the differential equation of the path, viz. x 3 + y* ' whence a 2 (x 2 +y 2 ) = 4x 2 y 2 , a being an arbitrary constant, or, transforming to polar co-ordinates, -sk (3) Also since x y-y%=-r~> the vortex moves as if under a centre of force at the origin. This force is repulsive, and its law is that of the inverse cube*. 156. If we write, as in Chapter IV., z = oc + iy, w = (f> + iyjr, (1) the potential- and stream -functions due to an infinite row of equidistant vortices, each of strength k, whose co-ordinates are (0,0), (±a, 0), (±2o, 0), ..., will be given by the formula ifc , . irz . w = ^logsm-; (2) cf. Art. 64, 4°. This makes dw %k , irz /ox u — iv— — 5-= — s- cot — , (6) dz 2a a whence k sinh {1iry\a) ___ k sin(27r#/a) 2a cosh (27n//a) — cos (2wx/a) ' 2a cosh (2iry/a) — cos (27nc/a) ' (4) * Greenhill, "On Plane Vortex- Motion," Quart. Journ. Math. xv. 10 (1878); Grobli, Die Bewegung paralleler geradliniger Wirbelfdden, Zurich, 1877. These papers contain other in- teresting examples of rectilinear vortex-systems. The case of a system of equal and parallel vortices whose intersections with the plane xy are the angular points of a regular polygon was treated by J. J. Thomson in his Motion of Vortex Rings, pp. 94.... He finds that the configura- tion is stable if, and only if, the number of vortices does not exceed six. For some further references as to special problems see Hicks, Brit. Ass. Rep. 1882, pp. 41...; Love, I.e. ante p. 192. An ingenious method of transforming plane problems in vortex-motion was given by Kouth, "Some Applications of Conjugate Functions," Proc. Lond. Math. Soc. xii. 73 (1881). 155-156] Rows of Vortices 225 These expressions make w= + \ic\a, v = 0, for y = ± oo ; the row of vortices is in fact, as regards distant points, equivalent to a vortex-sheet of uniform strength x/a (Art. 151). The diagram shews the arrangement of the stream-lines. It follows easily that if there are two parallel rows of equidistant vortices, symmetrical with respect to the plane y = 0, the strengths being k for the upper and — k for the lower row, as indicated on the next page, the whole system will advance with a uniform velocity £7 = ^coth— , (5) where b is the distance between the two rows. The mean velocity in the plane of symmetry is KJa. The velocity at a distance outside the two rows tends to the limit 0. If the arrangement be modified so that each vortex in one row is opposite the centre of the interval between two consecutive vortices in the other row, as shewn on p. 228, the general velocity of advance is F =^ tanh ? w The mean velocity in the medial plane is again k/cl. The stability of these various arrangements has been discussed by von Karman*. Taking first the case of the single row, let us suppose the vortex whose undisturbed co-ordinates are (ma, 0) to be displaced to the point (ma + x m , y m ). The formulae of Art. 154 give, for the motion of the vortex initially at the origin, dxQ = k yo-y m dy Q ^ < x -x m -ma dt 2rr m rj ' dt 2tt m rj ' { '> where r m 2 = (x -x m -ma)* + (y -y m ¥, (8) and the summation with respect to m includes all positive and negative integral values, zero being of course excluded. If we neglect terms of the second order in the displacements, we find dxp ^ k y<>-y m dyo = !L_ S I * ^ %Q-%m , q , dt 27ra 2 w m a ' dt 27ra 2 m m 2na 2 ^ m 2 {) * " Fhissigkeits- u. Luftwiderstand,' Phys. Zeitschr. xiii. 49 (1911); also Gdtt. Nachr. 1912, p. 547. The investigation is only given in outline in these papers; I have supplied various steps. 226 Vortex Motion [chap, vii The first term in the value of dy /dt is to be omitted as being independent of the disturbance*. Consider now a disturbance of the type x m = ae im *, y m = $e im *, (10) where <f> may be assumed to lie between and 2rr. If be small this has the character of an undulation of wave-length 2?ra/0. We find *--* S--* cm The arrangement is therefore unstable, the disturbance ultimately increasing as e kt . When the wave-length is large compared with a we have X = i K 0/a 2 , (13) approximately; cf. Art. 234. Proceeding next to the case of the symmetrical double row, the positions at time t of vortices in the upper and lower rows may be taken to be {ma+Ut + x m , \b+y m ), and (na+Ut + x n ', -%b+y n '\ respectively, where U denotes the general velocity of advance of the system, and the origin is in the plane of symmetry. (£> t> $ & <p (9 <9 <9 The component velocities of a vortex in the upper row, e.g. that for which ra = 0, due to the remaining vortices of the same row, will be given as before by (9), where the sum 2m -1 may be omitted. The components due to the vortex n of the lower row will be _^ b+yo-Vn < x -x n '-na 2tt r 2 ' 2tt r 2 where r n 2 = (x - x n ' -na) 2 + (y - y n ' + b) 2 . If we neglect terms of the second order in the disturbance we find, after a little reduction, 27r(dx \_ y Q -y m s b n 2 a 2 -b 2 2nab . , s .. .. + !(^w^-*° (1) 2tt dy _ ^ x -x m n 2 a 2 -b 2 k dt~ I m 2 a 2 + Z(n 2 a 2 + b 2 ) 2{X ° Xn} ,2^2TT2V2^0-yn), (15) n (n 2 a' + < where the summations with respect to n go from — oo to +00, including zero. The terms in (14) independent of the disturbance will cancel, since, by (5), T-r * 11 1"^ K „ b U=— coth — = — 2 a 2tt n n 2 a 2 + b 2 ' * In the summations the vortices are to be taken in pairs equidistant from the origin ; other- wise the result would be indeterminate. The investigation may be regarded as applying to the central portions of a long, but not infinitely long, row; the term referred to is then negligible. 156] Stability of doable rows If we now put where 0<$<27r, the equations take the form 2na 2 da k dt 2ira 2 dp k dt If we write, for shortness, the values of the coefficients are* -Ap-Ba'-Cp, = -Aa -Ca' + Bff. k=b/a, 2nke in< ^ _ . Jtt0 cosh h (ir — <f>) ir 2 sinh /ccf)\ l(ri*+¥j 2 ~ l \ sinh/br sinh 2 /br J ' ' {n 2 -k 2 )e in * 7r 2 cosh£<£ 7r<£sinh£(7r-<£) (7 = 2 » (n» + £*)» sinh 2 ^7r sinh yfc7r 227 .(16) .(17) .(18) .(19) .(20) .(21) To deduce the equations relating to the lower row we have merely to reverse the signs of < and b, and to interchange accented and unaccented letters. Hence 27ra 2 da = A(3?-Ba + C(3, 2tt« 2 d& k dt .(22) = Aa' +Ca+Bp. The formulae (17) and (22) are the equations of motion of the vortex-system in what may be called a normal mode of the disturbance. The solutions are of two types. In the first type we have a = a', /3=-/3', and therefore 2vra 2 da k dt = -Ba-(A-C)(3, The solution involves exponentials e xt , the values of X being given by 2tt« 2 B±J(A 2 -C 2 ). In the second type we have and therefore a=-a', /3 = /3', . 27ra 2 da „ . . ~. n ---=Ba-(A + C)P, 27ra 2 dp k dt The corresponding values of X are given by 27ra 2 = -(A-C)a+Bp. \ = B±J(A 2 -C 2 ). •(23) .(24) .(25) .(26) .(27) .(28) The summations with respect to n can be derived from the Fourier expansion cosh k (ir - <p) 1 Jl 2&cos<£ 2fccos20 [ sinhfor ~^\k + ~VTW + ~2 r +k ir + '''\' 228 Vortex Motion [chap, vii Since B is a pure imaginary, whilst A and C are real, it is necessary for stability in each case that A 2 should not exceed C 2 for admissible values of <£. Now when $ = 7r we find ,4 + C=4,r 2 tanh 2 |/br, A-C=%7r 2 coth 2 %k7r (29) so that A 2 — C 2 is positive. We conclude that both types are unstable. Passing to the unsymmetrical case, we denote the positions of the displaced vortices by (ma+Vt + x m , $b+y m ), and ((n + %)a+ Vt + x n \ -£&+#»)» where V is given by (6). The requisite formulae are obtained by writing n + ^ for n in preceding results. d> § 3) <S The equations (17) and (22) will accordingly apply, provided* 1 _ fi im<f> ( n i 1A2 _ Z.2 _2 „ ^(2n + l)ke i( ^ n+ ^^ . ( ir(f> sinh h (tt - 0) 7r 2 sinh^i .__. i * = 2 f/„ i 1X2 . 7.2)2 =Z 1 Z3^ EZ + _U2 EZ ) \ dl ) {(» + *)* + ' {(w + ^) 2 -F}e f(w+ * ) ^ _ 7r 2 cosh^ 7T(frcosh/fc(7r-(fr) « {0 + i) 2 + £ 2 } 2 ~ cosh 2 **- cosh/br ( ' These values of A, B, C are to be substituted in (25) and (28). As in the former case it is necessary for stability that A 2 should not be greater than C 2 . Now when <£ = 7r, (7=0; hence A must also vanish, or cosh 2 £7r = 2, &7r = -8814, &/« = & = -281 (33) The configuration is therefore unstable unless the ratio of the interval between the two rows to the distance between consecutive vortices has precisely this value. To determine whether the arrangement is stable, under the above condition, for all values of <f> from to 2n, let us write for a moment kin - <fi)=x, kir=n, so that k 2 A = — \x 2 , k 2 C= \ (fix cosh fix cosh x - fi 2 sinh fi sinh x), (34) where x may range between ±fi. Since A is an even and C an odd function of x, it is sufficient for comparison of absolute values to suppose x positive. Hence, writing y — fi cosh fx cosh x - fi 2 sinh fi x, (35) x we have to ascertain whether this is positive for 0<x<fi. Since fi= '8814, cosh/x= x /2, sinh ii=l t y is positive for x = 0, and it evidently vanishes for x = fi. Again -j- = fi cosh /x sinh x + fx 2 sinh fi — — ^ ju 2 sinh/x 1, (36) QjCC 00 00 which is equal to - 1 for #=0, and vanishes for x = fi. Finally, d 2 y , , . sinh a? _ „ . , cosh# _ „ . , sinh x . -r~2 = fi cosh fi cosh # - ^ sinh p h2^ 2 sinh/* 2/x/sinh/Li — ^— , ...(37) CLOG 00 0G Ob * The summations with respect to n can be derived from the expansion sinh k {tt - <f>) _ 2 jftcos^0 _ &cos|0 cosh kir <f>) _ 2 jk cos %<f> k cos f0 | 156-157] Stability of double rows 229 which is easily seen to be positive for all values of x, since (tanh^)/^?<l. Hence as x increases from to /x, dy\dx is steadily increasing from - 1 to 0, and is therefore negative. Hence y steadily diminishes from its initial positive value to zero, and is therefore positive. We conclude that the configuration is definitely stable* except for x= ±/z, when <f> = or 27r, in which cases B=0, by (31), and therefore X = 0. Since the disturbed particles are then all in the same phase, the reason why the period of disturbance should be infinite is easily perceived. This unsymmetrical configuration is of special interest because it is exemplified in the trail of vortices which is often observed in the wake of a cylindrical body advancing through a fluid. This has suggested further researches. The effect of lateral rigid boundaries equidistant from the medial line on the stability of the configuration has been discussed by Rosenhead f. He finds that as the ratio a/k of the interval a between successive vortices in the same row to the distance h between the walls increases from zero to *815 the unsymmetrical arrangement is stable only for a definite value of 6/a, which decreases continuously from -281 to '256. But when a/h> *815 there is stability for a certain range of values of b/a. And when a/h>l'4l9 the configuration is stable for all values of b/a. The symmetrical configuration, on the other hand, is always unstable. 157. When, as in the case of a vortex-pair, or a system of vortex-pairs, the algebraic sum of the strengths of all the vortices is zero, we may work out a theory of the 'impulse,' in two dimensions, analogous to that given in Arts. 119, 152 for the case of a finite vortex -system. The detailed examination of this must be left to the reader. If P, Q denote the components of the impulse parallel to x and y, and N its moment about Oz, all reckoned per unit depth of the fluid parallel to z, it will be found that P = ptfy£dxdy, Q = -pffx^dxdy jy N=-%pJS(x* + f)Sdxdy. J For instance, in the case of a single vortex-pair, the strengths of the two vortices being ± k, and their distance apart c, the impulse is pice, in a line bisecting c at right angles. The constancy of the impulse gives %kx = const., Xfcy — const., j %/c (x 2 + y 2 ) = const. ) It may also be shewn that the energy of the motion in the present case is given by T=-\ 9 \\^dxdy = -\plK^ (3) When 2k is not zero, the energy and the moment of the impulse are both infinite, as may be easily verified in the case of a single rectilinear vortex. * This is stated without proof by Karman. t Phil. Trans. A, ccviii. 275 (1929). See also Glauert, Proc. Roy. Soc. A, cxx. 34 (1928). 230 Vortex Motion [chap, vii The theory of a system of isolated rectilinear vortices has been put in a very elegant form by Kirchhoff *. Denoting the positions of the centres of the respective vortices by (# l5 yi), (x 2 , y 2 ), ••• and their strengths by k 1} k 2 , • ••> it is evident from Art. 154 that we may write Kl dt dy x ' * l dt dccy ' dx 1:= _dW fy2_<^ , l (4) * 2 dt ~ 83/2 ' 2 dt ~ dx 2 where w =k~ ^ K i K a^°S r i2i (5) Air if r 12 denote the distance between the vortices k 1? k 2 . Since TF depends only on the relative configuration of the vortices, its value is unaltered when #1, # 2 > ••• are increased by the same amount, whence 23^/8^=0, and, in the same way, 28TF/8y! = 0. This gives the first two of equations (2), but the proof is not now limited to the case of 2k =0. The argument is in fact substantially the same as in Art. 154. Again, we obtain from (4) / dx dy\ / dW dW\ or if we introduce polar co-ordinates (r u $i\ (r 2 , 2 ), ... for the several vortices, 2Kr di=-*w (6) Since W is unaltered by a rotation of the axes of co-ordinates in their own plane about the origin, we have 23 WJ 80=0, whence 2«r 2 =const., (7) which agrees with the third of equations (2), but is free from the restriction there implied. An additional integral of (4) is obtained as follows. We have s "{ x dt y Tt)-*\ x dx +y ty ) ~3-»? « If every r be increased in the ratio 1 + e, where € is infinitesimal, the increment of W is equal to 2er . 8 W/dr. But since the new configuration of the vortex-system is geometrically similar to the former one, the mutual distances r 12 are altered in the same ratio 1 + e, and therefore, from (5), the increment of W is e/27r.2Ki/c 2 . Hence (8) may be written in the form ^S-c^- (9) 158. The preceding results are independent of the form of the sections of the vortices, so long as the dimensions of these sections are small compared with the mutual distances of the vortices themselves. The simplest case is when the sections are circular, and it is of interest to inquire whether this form is stable. This question has been examined by Kelvin f. * Mechanik, c. xx. t Sir W. Thomson, "On the Vibrations of a Columnar Vortex," Phil. Mag. (5), x. 155 (1880) [Papers, iv. 152]. 157-158] Stability of a Columnar Vortex 231 When the disturbance is in two dimensions only, the calculations are very simple. Let us suppose, as in Art. 27, that the space within a circle r = a, having the centre as origin, is occupied by fluid having a uniform vorticity co, and that this is surrounded by fluid moving irrotationally. If the motion be continuous at this circle we have, for r<a, *=-i*(« 2 -r 2 ), (i) while for r>a, >//•= — \ coa 2 log a/r (2) To examine the effect of a slight irrotational disturbance, we assume, for r<a, + = - i a) (a 2 - r 2 ) + A - 8 cos (s0 - <r*),l I (3) and, for r > a, yjr = - ^ coa 2 log - + A — cos (s0 - at\ where s is integral, and a is to be determined. The constant A must have the same value in these two expressions, since the radial component of the velocity, —dtyjrdB, must be continuous at the boundary of the vortex, for which r=a, approximately. Assuming for the equation to this boundary r= a + a cos (s6- at), (4) we have still to express that the transverse component (dy^/dr) of the velocity is continuous. This gives A -coa 2 A k cor + s — cos (s6 - at) =- 5 — cos (s0 - at). a s r a v Substituting from (4), and neglecting the square of o, we find coo= —2sAja (5) So far the work is purely kinematical; the dynamical theorem that the vortex-lines move with the fluid shews that the normal velocity of a particle on the boundary must be equal to that of the boundary itself. This condition gives dr _ d\jr dyj/ dr di~ ~ rd6~ dr rd6' where r has the value (4), or A , i sa fa\ aa = s — + icoa.— (o) a a Eliminating the ratio A /a between (5) and (6) we find o-=4(s-l)co (7) Hence the disturbance represented by the plane harmonics in (3) consists of a system of corrugations travelling round the circumference of the vortex with an angular velocity als = (s-l)ls.%<o (8) This is the angular velocity in space ; relative to the rotating fluid the angular velocity is a/s— |co= -^co/s, (9) the direction being opposite to that of the rotation. When 5=2, the disturbed section is an ellipse which rotates about its centre with angular velocity £co. The three-dimensional oscillations of an isolated columnar vortex-filament have also been discussed by Kelvin in the paper cited. The columnar form is found to be stable for disturbances of a general character. In a recent paper Rosenhead* has examined the stability of the Karman unsymmetrical arrangement when the cross-sections are of finite area. The conclusion is that there is stability for strictly two-dimensional disturbances, but instability for sinusoidal longitudinal deformations, whose wave-length bears less than a certain ratio to the diameter. * Proc. Roy. Soc. A, cxxvii. 590 (1930). 232 Vortex Motion [chap, vti 159. The particular case of a two dimensional elliptic disturbance can be solved without approximation as follows*. Let us suppose that the space within the ellipse is occupied by liquid having a uniform vorticity a>, whilst the surrounding fluid is moving irrotationally. It will appear that the conditions of the problem can all be satisfied if we imagine the elliptic boundary to rotate, without change of shape, with a constant angular velocity (n, say), to be determined. The formula for the external space can be at once written down from Art. 72, 4° ; viz. we have yj, = %n (a + bfe- 2 * cos 2r)+%a>ab£, (2) where £, r\ now denote the elliptic co-ordinates of Art. 71, 3°, and the cyclic constant k has been put = 7ruba>. The value of \js for the internal space has to satisfy H + p=°" ( 3 > with the boundary-condition — ^ + -A = —ny> -^ + nx . ^ (4) These conditions are both fulfilled by + = ia(Ax* + Btf*), (5) provided A + B = l, Aa 2 - Bb 2 = - (a 2 -b 2 ) (6) CO It remains to express that there is no tangential slipping at the boundary of the vortex; i.e. that the values of 8^/3| obtained from (2) and (5) there coincide. Putting x=c cosh ^ cost), y = csinh£sinr7, where c=*J(a 2 -b 2 ), differentiating, and equating coeffi- cients of cos 2?7, we obtain the additional condition -%n(a + b) 2 e- 2 *=Uc 2 (A -B)cosh£sinh$, where £ is the parameter of the ellipse (1). This is equivalent to A ^ B —i.±g, (7 , a) ab y ' since, at points of the ellipse, cosh£ = a/c, sinh£ = 6/c. Combined with (6) this gives Aa — Bb— -, (8) and n= (^bf" < 9) When a = 6, this agrees with our former approximate result. The component velocities x, y of a particle of the vortex relative to the principal axes of the ellipse are given by whence we find -=—n\. y—n- (10) a 6' b a v ' * Kirchhoff, Mechanik, c. xx.; Basset, Hydrodynamics, ii. 41. 159-159 a] Elliptic Vortex 233 Integrating, we find x — ka cos {nt + e), y = kbsin (nt+e), (11) where k, e are arbitrary constants, so that the relative paths of the particles are ellipses similar to the boundary of the vortex, described according to the harmonic law. If %', y' be the co-ordinates relative to axes fixed in space, we find x = x cos nt - y sin nt = \h (a + b) cos (2nt + e) + ^k(a-b) cos e, •(12) y' = x sin nt + y cos nt = \ k (a + b) sin (2nt + e) - %k {a - b) sin e. The absolute paths are therefore circles described with angular velocity 2n 159 a. The motion of a solid in a liquid endowed with vorticity is a problem of considerable interest, but is unfortunately not very tractable. The only exception is when the motion is two-dimensional, and the vorticity uniform. Let x , y be the co-ordinates, relative to fixed axes, of a point C of the (cylindrical) solid ; let x, y be the co-ordinates of any point of the fluid relative to parallel axes through C, and let {u, v) be the velocity relative to C. We have then du .. du dv „ 1 dp dv .. du dv „ 1 dp .(1) cf. Arts. 12 (3) and 146 (5). Since u-- d -± v- d ± (2) and £ is constant, it appears that dujdt and dv/dt are the derivatives with respect to x and y, respectively, of a certain function of x, y, t. Denoting this function by - dcfr/dt, we have dx\dt)~ dt dy\dt)j dy\dt)~ dt dx\dt)> {) which are the conditions that -j- (<p + iyj/) should be a function of the complex variable x + iy. This consideration determines dcj)/dt when the form of y\r is known t. The equations (1) now give ^ d ^-(x x+y y)-^ + C^, (4) where q2 = u 2 + v 2 (5) We proceed to apply these results to some cases of motion of a circular cylinder. The point C is naturally taken on its axis. Let us suppose in the first instance that the undisturbed motion of the fluid consists of a uniform rotation co about the origin, so that £=2o>. The stream-function for the motion relative to a moving point (x , y ) is then ^o = i a) {(x + xf + (y +y) 2 } + x y - y x = \ (or 2 + cor (x cos 6 + y sin 6) + ^ a> (x 2 +y 2 )+r(x sin 0-y o cos 6), (6) * For further researches in this connection see Hill, "On the Motion of Fluid part of which is moving rotationally and part irrotationally," Phil. Trans. 1884; Love, «'On the Stability of certain Vortex Motions," Proc. Lond. Math. Soc. (1) xxv. 18 (1893). f Cf. Proudman, "On the Motion of Solids in a Liquid possessing Vorticity," Proc. Roy. Soc. A, xcii. 408 (1916). 234 Vortex Motion [chap, vii where we have introduced polar co-ordinates relative to C. The relative stream-function for the disturbed motion will be ^=^<or 2 + a> (r j (x o cos0+y o sm0) + %a>(xQ Z +yQ 2 ) + (r — - J (ir sin^-^ cos^). (7) For this satisfies V 1 2 >Jr = 2<o; it makes \jr = const, for r=a; and it agrees with (6) for r=oo . Hence 37 = G> \ r ) (%QCOsd+y sm8) + \ r ) (x aind-i/ cos6), (8) and therefore ^r= -a, ( r -\ — J (ir o sin0-y o cos0) + lr-\ — J (#ocos0+j/ o sin0), (9) terms independent of r and 6 being omitted. Again we have, for r=a, -^ =0, jf- = coa + 2co (x cos 6 +y sin 0) + 2 (i; sin - $ cos 0), and therefore ^ 2 =2cD 2 a(^ o cos0-|-yo sm ^) + 2(aa(Aosin0-^oCos0) + etc., (10) where terms are omitted which will contribute nothing to the resultant force on the cylinder. Substituting in (4) we find, for r=a, V) - = a(x o cos0+y o sin 6) - 4a>a (ir sin — $ cos#) - 2o> 2 a (# cos# +y sin 0) + etc. . ..(11) The component forces on the cylinder, due to fluid pressure, are therefore* /lit p cos a d0 = - M' (x Q + 4g># - 2g> 2 # ), /2jt p sin ad3= —M' (y - 4© x — 2<» 2 y ), where M' = npa 2 . Hence if M be the mass per unit length of the cylinder itself, the equations of motion are H fii/ — 4a>x — 2<a 2 y = Y\M',) where /u=l + Jf/Jf , and the zero suffixes have been omitted as no longer necessary. If we write z=x+iy, these equations are equivalent to f£-4:iG>z-2<o 2 z=(X+iY)/M' (14) To ascertain the free motion, when X=0, F=0, we assume that z oc e imo,t , and find /iwi 2 -4m + 2 = (15) If /n<2, i.e. if the mass of the cylinder is less than that of the fluid which it displaces, the values of m are real, and the solution has the form z^AeWi^ + Be™*"*, (16) where m u m 2 are positive. This represents motion in a 'direct 5 epicyclic. As special cases circular paths are possible, and are stable. If on the other hand /x>2, the values of m are complex, and the solution takes the form z=(Ae* t + Be-« t )e i f i \ (17) the ultimate path being an equiangular spiral. If /x = 2, we have (m- 1) 2 =0, and z=(A+Bt)e i » t (18) * Cf. G. I. Taylor, "Motion of Solids in Fluids when the Flow is not Irroiational, " Proc. Boy. Soc. A, xciii. 99 (1916). .(12) fix + 4a>y - 2<o 2 x = X\ M',) .(13) 159 a] Cylinder in Rotating Fluid 235 Hence, although it is possible as we should expect for a cylinder having the same mean density as the fluid to revolve with the latter in a circular path, this motion is unstable. If there is a radial force whose direction revolves with the fluid, say X+iY=R*~, (19) the equation (14) is satisfied, when /* = 2, by z = re i <* t , (20) provided r=\R\M' (21) The cylinder can therefore move, relatively to the rotating fluid, along a radius*, but this motion, again, must be classed as unstable t. Let us next suppose that the fluid when undisturbed is in laminar motion parallel to Ox, with constant vorticity 2<o, the stream -function being V'o=a>(yo+.y) 2 =W 2 ( l ~ cos 2 ^) + 2a #<>r sintf + a>y 2 (22) In the disturbed motion relative to the cylinder ^ = ^o>r 2 -|o> (r 2 -\\ cos28 + 2a>y (r- < —\ sm0 + ny o 2 + (r- — J (x sin 6 - y Q cos 6). (23) Hence -^- = 2(oy (r- — J sin 6+(r J (i? sin 0-y o cos0), (24) the terms independent of r and 6 being omitted. We write therefore ^ = 2coy o (r + ^cos6 + (r+^)(x^co&6+i/o8in0^ (25) For r = a we have from (23) -=^ = 0, ~= -<oa + 4a>a sin 2 0+4e«>yo sin + 2 (x sin — y cos 0), (26) and therefore \ q 2 = — 4o) 2 ay sin 6 — 2a>a (x sin 6 - y cos 0) + 1 6o> 2 ay sin 3 + 8a>ay (#o sin 2 — i/ Q sin 2 cos 0) + etc., (27) those terms only being retained which will contribute to the resultant force on the cylinder. Substituting in (4) we find, for r = a, P —=a(x cos +y sin 0) + 2aax (sin — 4 sin 3 0) + 2o>ay (cos + 4 sin 2 cos 0) + 4o> 2 ay (sin - 4 sin 3 0) + etc (28) /•2tt Hence % — \ P cos 0ad0= —M' (x + 4g># )> / 2tt jd sin 0ad0 = —M' (i/ — 4a>x — 8o> 2 y ). The equations of motion of the cylinder are therefore, omitting the suffixes, fix + 4a>y = X/M'j (29) lii/-4a>x-8 > 2 y=r/M'.! We notice that the cylinder can remain at relative rest subject to a force T= -8 a > 2 M'y=4 >M'U=2K P U, (31) * Cf. Taylor, I.e. t Some cases of motion of a sphere in rotating fluid have been studied by Proudman, I.e.; S. F. Grace, Proc. Roy. Soc. A, cii. 89 (1922); and Taylor, Proc. Roy. Soc. A, cii. 180 (1922). % Cf. Taylor, I.e. 236 Vortex Motion [chap, vii where U{= - 2a>i/) is the velocity of the undisturbed stream at the level of the centre, and k ( = 27r« 2 o>) is the circulation immediately round the cylinder. This result may be con- trasted with Art. 69 (6). It is easily found from (30) that, if /i<2, the path when there are no extraneous forces is a trochoid whose general direction of advance is parallel to the stream. 160. It was pointed out in Art. 80 that the motion of an incompressible fluid in a curved stratum of small and uniform thickness is completely denned by a stream-function yjr, so that any kinematical problem of this kind may be transformed by projection into one relating to a plane stratum. If, further, the projection be 'orthomorphic,' the kinetic energy of corresponding portions of liquid, and the circulations in corresponding circuits, are the same in the two motions. The latter statement shews that vortices transform into vortices of equal strengths. It follows at once from Art. 145 that in the case of a closed simply-connected surface the algebraic sum of the strengths of all the vortices present is zero. We may apply this to motion in a spherical stratum. The simplest case is that of a pair of isolated vortices situated at antipodal points ; the stream-lines are then parallel small circles, the velocity varying inversely as the radius of the circle. For a vortex-pair situate at any two points A, B, the stream-lines are coaxal circles as in Art. 80. It is easily found by the method of stereographic projection that the velocity at any point P is the resultant of two velocities k fair a . cot ^0 1 and ufa-ira . cot jj0 2 > perpendicular respectively to the great-circle arcs AP, BP, where U 2 dbriote the lengths of these arcs, a the radius of the sphere, and ±< the strengths of the vortices. The centre* (see Art. 154) of either vortex moves perpendicular to AB with a velocity k/27t« . cot \AB. The two vortices therefore describe parallel and equal small circles, remaining at a constant distance from each other. Circular Vortices. 161. Let us next take the case where all the vortices present in the liquid (supposed unlimited as before) are circular, having the axis of x as a common axis. Let to- denote the distance of any point P from this axis, v the velocity in the direction of ot, and a> the resultant vorticity at P. It is evident that u, v, co are functions of x, zr only. Under these circumstances there exists a stream-function yjr, defined as in Art. 94, viz. we have m= _i^ ¥ _i£ tar c-gt ts ox whence a> = -- — =.- ( *+* JL\ (2) ox ost nr\oar o-sr m otz ) It is easily seen from the expressions (7) of Art. 148 that the vector (F, G, H) will under the present conditions be everywhere perpendicular to * To prevent possible misconception it may be remarked that the centres of corresponding vortices are not necessarily corresponding points. The paths of these centres are therefore not in general projective. i59a-i6i] Circular Vortices 237 the axis of x and the radius -&. If we denote its magnitude by S, the flux through the circle (x, m) will be 2ir^S, whence ^ = -*tS (3) To find the value of i/r at (x, ©■) due to a single vortex-filament of circulation tc, whose co-ordinates are x', w', we note that the element which makes an angle 6 with the direction of S may be denoted by vt'hd, and therefore by Art. 149 (1) ic&w' f 27r cos 6 ■ t— rf ^— 5tJ — m, w where r = {(x- x'f + w 2 + w' 2 - 2*nsr' cos 0}i (5) If we denote by r 1} r 2 the least and greatest distances, respectively, of the point P from the vortex, viz. r 2 = (x- x') 2 + O - m') 2 , r 2 2 = (x - x'f + (# + OT ') 2 , (6) we have r 2 = j\ 2 cos 2 J + ?' 2 2 sin 2 J #, 4<*tot-' cos 6 — r ± 2 + r 2 2 — 2r 2 , (7) and therefore i K (n 2 + r 2 2 ) J V(ri 2 cos 2 ^ + r 2 2 sin 2 ^) 2 f V (n 2 cos 2 i (9 + r 2 2 sin 2 § (9) dd Jo (8) The integrals are of the types met with in the theory of the ' arithmetico- geometrical mean.'* In the ordinary, less symmetrical, notation of 'complete' elliptic integrals we have t = -^(-')*{g-^ lW -|^ W }, (9) provided y-l-gg ^- ^ ^ , x2 (10) 4gg' r 2 2 (0 - x') 2 + (tsr + t*') 2 The value of -^ at any assigned point can therefore be computed with the help of Legendre's tables. A neater expression may be obtained by means of 'Landen's trans- formation,' f viz. f = -£(r l + r 2 ){F 1 (X)-E 1 (X)}, (11) provided x^HZl* (12) * ri + ri v The forms of the stream-lines corresponding to equidistant values of yjs are shewn on the next page. They are traced by a method devised by Maxwell, to whom the formula (11) is also due J. * See Cayley, Elliptic Functions, Cambridge, 1876, c. xiii. t See Cayley, I.e. J Electricity and Magnetism, Arts. 704, 705. See also Minchin, Phil. Mag. (5), xxxv. (1893); Nagaoka, Phil. Mag. (6), vi. (1903). 238 Vortex Motion CHAP. VII Expressions for the velocity-potential and the stream-function can also be obtained in the fcrm of definite integrals involving Bessel's Functions. X'- Thus, supposing the vortex to occupy the position of the circle x = 0, tzr=a, it is evident that the portions of the positive side of the plane x = which lie within and without this- 161-162] Stream-lines of a Vortex Ring 239 circle constitute two distinct equipotential surfaces. Hence, assuming that we have $ = %< for # = 0, w<a, and <£=0 for #=0, w>a, we obtain from Art. 102 (2) = i K a f ^ e- kx J Q {kw)J l {ka)dk, (13) and therefore, in accordance with Art. 100 (5), ty=-% K aw J e-te^^Jiikcfidk (14) These formulae relate of course to the region x >0*. It was shewn in Art. 150 that the value of <p is that due to a system of double sources distributed with uniform density k over the interior of the circle. The values of cf> and -v//- for a uniform distribution of simple sources over the same area have been given in Art. 102 (11). The above formulae (13) and (14) can thence be derived by differentiating with respect to x, and adjusting the constant factor f. 162. The energy of any system of circular vortices having the axis of x as a common axis, is T= irp \(u 2 + v 2 ) ■urdxd'n = 7rp I j(v ~ — u~-j dxdts = — irp I jyfrcodxd'OT = — 7rpX«^, (1) by a partial integration, the integrated terms vanishing at the limits. We have here used k to denote the strength coSxSzz of an elementary vortex-filament. Again the formula (7) of Art. 153 becomes^ T = 27T/o//(OTtt — xv) mwdxdy— 2TrpX/cur {tzu —xv) (2) The impulse of the system obviously reduces to a force along Ox. By Art. 152 (6), P = J p fj \y% ' — zrj) dxdydz = irp Jj^codxd'ST = rrpX/m* (3) If we introduce the symbols x , w denned by the equations 2/csr 2 # 9 2/eor 2 ^o=^ 2, W = ~^—, (4) these determine a circle whose position evidently depends on the strengths and the configuration of the vortices, and not on the position of the origin on the axis of symmetry. It may be called the 'circular axis' of the whole system of vortex-rings. * The formula for \j/ occurs in Basset, Hydrodynamics, ii. 93. See also Nagaoka, I.e. t Other expressions for <f> and x}/ can be obtained in terms of zonal spherical harmonics. Thus the value of <p is given in Thomson and Tait, Art. 546 ; and that of \f/ can be deduced by the formulae (11), (12) of Art. 95 ante. The elliptic-integral forms are however the most useful for purposes of interpretation. { At any point in the plane 2 = we have y = w, £ = 0, y = 0, f=£w, v = v; the rest follows by symmetry. 240 Vortex Motion [chap, vii Since k is constant for each vortex, the constancy of the impulse shews, by (3) and (4), that the circular axis remains constant in radius. To find its motion parallel to x, we have, from (4), Ik . ctq 2 . -jj — It/cn 2 -=r + 2%/cgfx -77 = Xkhf (vtu + 2xv) (5) With the help of (2) this can be put in the form 2/C.OTo 2 . -37 = 2^" + 32* (x - COq) VTV, (6) where the added term vanishes, since ^kutv = on account of the constancy of the mean radius (txr ). 163. Let us now consider, in particular, the case of an isolated vortex-ring the dimensions of whose cross-section are small compared with the radius (-5t ). It has been shown that ♦-^//K^)-*^)}^^^^ « where r 1} r 2 are defined by Art. 161 (6). For points (x, m) in or near the substance of the vortex, the ratio r\\r% is small, and the modulus (X) of the elliptic integrals is accordingly nearly equal to unity. We then have ^ 1 (X) = |logi|, &W-1, (2) approximately*, where X' denotes the complementary modulus, viz. x ^ 1 - x, -5S3?' (3 > or X' 2 = 5ri/r 2 , nearly. Hence at points within the substance of the vortex the value of yjr is of the order /csy log (wo/e), where e is a small linear magnitude comparable with the dimensions of the section. The velocities at such points, depending (Art. 94) on the differential coefficients of yfr, will be of the order x/e. We can now estimate the magnitude of the velocity dx /dt of translation of the vortex-ring. By Art. 162 (1), T is of the order ptc 2 Gr \og (sr /e), and v is, as we have seen, of the order /e/e; whilst x — x is of course of the order e. Hence the second term on the right-hand side of the formula (6) of the preceding Art. is, in the present case, small compared with the first, and the velocity of translation of the ring is of the order k/vfq . log (w /e), and approxi- mately constant. An isolated vortex-ring moves then, without sensible change of size, parallel to its rectilinear axis with nearly constant velocity. This velocity is small compared with that of the fluid in the immediate neighbourhood of the circular axis, but may be greater or less than J/c/oto, the velocity of the fluid at the centre of the ring, with which it agrees in direction. * See Cayley, Elliptic Functions, Arts. 72, 77; and Maxwell, I.e. 162-163] Speed of a Vortex-Ring 241 For the case of a circular section more definite results can be obtained as follows. If we neglect the variations of w and a> over the section, the formulae (1) and (2) give *=-£-.//(i°g 8 f°-*)<w or, if we introduce polar co-ordinates (s, %) in the plane of the section, ♦--E-tf/r^-")'**' (4) where a is the radius of the section. Now I ' log r x dx=f * log {s 2 + s' 2 - 2ss' cos ( x - xjfi d Xi and this definite integral is known to be equal to 2tt logs', or 2?r log s, according as s'^s. Hence, for points within the section, *- - o. CTo f (log §p - 2) s'M - vwJ^ (log ^r»-a) *' *' = -Ja^a* {log^-f-^} (5) The only variable part of this is the term ^a>w s 2 ; this shews that to our order of approxi- mation the stream-lines within the section are concentric circles, the velocity at a distance s from the centre being ^a>s. Substituting in Art. 162 (1) we find 277-p The last term in Art. 162 (6) is equivalent to h=-H:!>^-£^-*} (e) f ot co2k (x-Xq) 2 . In our present notation, where k denotes the strength of the whole vortex, this is equal to f K 2 ar /7r. Hence the formula for the velocity of translation of the vortex becomes* dx < L 8tsr ] •(7) The vortex-ring carries with it a certain body of irrotationally moving fluid in its career; cf. Art. 155, 2°. According to the formula (7) the velocity of translation of the vortex will be equal to the velocity of the fluid at its centre when zzr /a = 86, about. The accompanying mass will be ring-shaped or not, according as or /a exceeds or falls short of this critical value. The ratio of the fluid velocity at the periphery of the vortex to the velocity at the centre of the ring is 2a>ai37 //c, or -as^na. For a = T JoZ<r , this is equal to 32, about. The conditions under which a vortex-ring of finite section and uniform vorticity can travel unchanged have been investigated by Lichtenstein f . The shape of the section, when small, is found to be approximately elliptic, with the minor axis in the direction of translation. He has also discussed the analogous question relating to a vortex-pair (Art. 155). * This result was given without proof by Sir W. Thomson in an appendix to a translation of Helmholtz' paper, Phil. Mag. (4), xxxiii. 511 (1867) [Papers, iv. 67]. It was verified by Hicks, Phil. Trans. A, clxxvi. 756 (1885); see also Gray, "Notes on Hydrodynamics," Phil. Mag. (6), xxviii. 13 (1914). t Math. Zeitsch. xxiii. 89, 310 (1925). See also his Grundlagen der Hydrodynamik, Berlin, 1829. 242 Vortex Motion [chap, vn 164. If we have any number of circular vortex-rings, coaxal or not, the motion of any one of these may be conceived as made up of two parts, one due to the ring itself, the other due to the influence of the remaining rings. The preceding considerations shew that the second part is insignificant com- pared with the first, except when two or more rings approach within a very small distance of one another. Hence each ring will move, without sensible change of shape or size, with nearly uniform velocity in the direction of its rectilinear axis, until it passes within a short distance of a second ring. A general notion of the result of the encounter of two rings may, in par- ticular cases, be gathered from the result given in Art. 149 (3). Thus, let us suppose that we have two circular vortices having the same rectilinear axis. If the sense of the rotation be the same for both, the two rings will advance, on the whole, in the same direction. One effect of their mutual influence will be to increase the radius of the one in front, and to contract the radius of the one in the rear. If the radius of the one in front becomes larger than that of the one in the rear, the motion of the former ring will be retarded, and that of the latter accelerated. Hence if the conditions as to relative size and strength of the two rings be favourable, it may happen that the second ring will overtake and pass through the first. The parts played by the two rings will then be reversed ; the one which is now in the rear will in turn overtake and pass through the other, and so on, the rings alternately passing one through the other*. If the rotations be opposite, and such that the rings approach one another, the mutual influence will be to enlarge the radius of each. If the two rings be moreover equal in size and strength, the velocity of approach will continually diminish. In this case the motion at all points of the plane which is parallel to the two rings, and half-way between them, is tangential to this plane. We may therefore, if we please, regard the plane as a fixed boundary to the fluid on either side, and so obtain the case of a single vortex-ring moving directly towards a fixed rigid wall. The foregoing remarks are taken from Helm hoi tz' paper. He adds, in conclusion, that the mutual influence of vortex-rings may easily be studied experimentally in the case of the (roughly) semicircular rings produced by drawing rapidly the point of a spoon for a short space through the surface of a liquid, the spots where the vortex-filaments meet the surface being marked by dimples. (Cf. Art. 27.) The method of experimental illustration by means of smoke-rings | is too well-known to need description here. A beautiful * Cf. Hicks, "On the Mutual Threading of Vortex Kings," Proc. Rqy. Soc. A, cii. Ill (1922). The corresponding case in two dimensions was worked out and illustrated graphically by Grobli, I.e. ante p. 224; see also Love, "On the Motion of Paired Vortices with a Common Axis," Proc. Lond. Math. Soc. xxv. 185 (1894), and Hicks, I.e. f Reusch, " Ueber Ringbildung der Flussigkeiten," Pogg. Ann. ex. (1860); Tait, Recen Advances in Physical Science, London, 1876, c. xii. 164-165] Mutual Influence of Vortex-Rings 243 variation of the experiment consists in forming the rings in water, the sub- stance of the vortices being coloured*. The motion of a vortex-ring in a fluid limited (whether internally or externally) by a fixed spherical surface, in the case where the rectilinear axis of the ring passes through the centre of the sphere, has been investigated by Lewis f, by the method of 'images.' The following simplified proof is due to Larmor %. The vortex-ring is equivalent (Art. 150) to a spherical sheet of double-sources of uniform density, concentric with the fixed sphere. The ' image ' of this sheet will, by Art. 96, be another uniform concentric double-sheet, which is, again, equivalent to a vortex-ring coaxal with the first. It easily follows from the Art. last cited that the strengths (k, k') and the radii (sr, g/) of the vortex-ring and its image are connected by the relation kw% + k'w'% = (1) The argument obviously applies to the case of a re-entrant vortex of any form, provided it lie on a sphere concentric with the boundary. The interest attaching to Karman's stable configuration of a system of line- vortices of small section (Art. 156) has led to the discussion of analogous arrangements in three dimensions. Considering, in the first instance, a procession of equal vortex-rings of infinitesimal section, spaced at equal intervals with a common axis, Levi and Forsdyke§ find that the arrangement is unstable for a type of disturbance in which the radii and the intervals vary simultaneously, the rings remaining accurately plane and circular. On the other hand, provided the ratio of the interval between successive rings to the common radius exceeds 1*20, periodic vibrations about the circular form are possible, of types discussed by J. J. Thomson and Dyson in the case of an isolated ring||. They examine next the case of a helical vortex 1F. If undisturbed this will have a certain angular velocity about its axis, and a certain velocity of advance. They find that there is stability if, and only if, the pitch of the helix exceeds 03. The Conditions for Steady Motion. 165. In steady motion, i.e. when ^ = ^ = ^ = dt ' dt ' dt u ' the equations (2) of Art. 6 may be written du dv t dw , . dfl ldp .... w d-x +v d- x +w te-w- wr >^-te-- P £ (1) * Keynolds, "On the Kesistance encountered by Vortex Kings &c," Brit. Ass. Rep. 1876; Nature, xiv. 477. f " On the Images of Vortices in a Spherical Vessel," Quart. Journ. Math. xvi. 338 (1879). X "Electro-magnetic and other Images in Spheres and Planes," Quart. Journ. Math, xxiii. 94 (1889). § Proc. Roy. Soc. A, cxiv. 594; A, cxvi. 352 (1927). || For references see p. 246. H Proc. Roy. Soc. A, cxx. 670 (1928). 244 Vortex Motion [chap, vii Hence, if as in Art. 146 we put x' = /f + i2 2 +n. (2) we have ^—v^ — wr^ ^r- = wi; — u£, -~—ur]-v^ (3) It follows that u-% +v£- + w-£- = 0, ox oy oz so that each of the surfaces tf = const, contains both stream-lines and vortex- lines. If further 8n denote an element of the normal at any point of such a surface, we have dy' ^ = qw sin/3, (4) where q is the current velocity, co the vorticity, and /3 the angle between the stream-line and the vortex-line at that point. Hence the conditions that a given state of motion of a fluid may be a possible state of steady motion are as follows. It must be possible to draw in the fluid an infinite system of surfaces each of which is covered by a network of stream -lines and vortex-lines, and the product qw sin @8n must be constant over each such surface, 8n denoting the length of the normal drawn to a con- secutive surface of the system *. These conditions may also be deduced from the considerations that the stream-lines are, in steady motion, the actual paths of the particles, that the product of the angular velocity into the cross-section is the same at all points of a vortex, and that this product is, for the same vortex, constant with regard to the time. The theorem that the function %', defined by (2), is constant over each surface of the above kind is an extension of that of Art. 21, where it was shewn that x is constant along a stream-line. The above conditions are satisfied identically in all cases of irrotational motion, provided of course the boundary-conditions be such as are consistent with the steady motion. In the motion of a liquid in two dimensions (xy) the product q8n is con- stant along a stream-line ; the conditions in question then reduce to this, that the vorticity £ must be constant along each stream-line, or, by Art. 59 (5), 3+^=/w. < 5 > where /(^) is an arbitrary function of yfr f. * See a paper "On the Conditions for Steady Motion of a Fluid," Proc. Lond. Math. Soc. (1) ix. 91 (1878). t Cf. Lagrange, Nouv. Mem. de VAcad. de Berlin, 1781 [Oeuvres, iv. 720]; and Stokes, "On the Steady Motion of Incompressible Fluids," Gamb. Trans, vii. (1842) [Papers, i. 15]. 165] Conditions for Steady Motion 245 This condition is satisfied in all cases of motion in concentric circles about the origin. Another obvious solution of (5) is + = \{Ax* + 2Bxy + Cy*\ (6) in which case the st eam-lines are similar and coaxal conies. The angular velocity at any point is -| (A + C), an 1 is therefore uniform. Again, if we put j (\jr) = - k 2 \js, where k is a constant, and transform to polar co-ordinates r, 0, we get which is satisfied (Art. 101) by ty = CJ s {kr) C08 \ sd (8) This gives various solutions consistent with a fixed circular boundary of radius w, the admissible values of k being determined by J s (ka)=0 (9) Suppose, for example, that in an unlimited mass of fluid the stream-function is + = CJi(kr)sm6, (10) within the circle r=a, whilst outside this circle we have f=u(r~\smd. .(11) These two values of \js agree for r=a, provided J x (ka)=0. Moreover, the tangential velocity at this circle will be continuous, provided the two values of d^jrjdr are equal, i.e. if W(ba) kJ (ka) y } If we now impress on everything a velocity U parallel to Ox, we get a species of cylindrical vortex travelling with velocity U through a liquid which is at rest at infinity. The smallest of the possible values of k is given by kaf-n- = 1'2197 ; the relative stream-lines inside the vortex are then given by the lower diagram on p. 288, provided the dotted circle be taken as the boundary (r=a). It is easily proved, by Art. 157 (I), that the 'impulse' of the vortex is represented by 2irpa 2 U. In the case of motion symmetrical about an axis (%), we have q . 27rtsrSn constant along a stream-line, -cr denoting as in Art. 94 the distance of any point from the axis of symmetry. The condition for steady motion then is that the ratio a/vr must be constant along any stream-line. Hence, if t/t be the stream-function, we must have, by Art. 161 (2), a^ + a^-- a — -VW> (is) where /(-v/r) denotes an arbitrary function of yfr*. An interesting example is furnished by Hill's ' Spherical Vortex f.' If we assume ^=4^37 2 (a 2 -r 2 ) (14) where r 2 =# 2 + zsr 2 , for all points within the sphere r=a, the formula (2) of Art. 161 makes 0)= —^A'STy so that the condition of steady motion is satisfied. Again it is evident, on reference to Arts. 96, 97, that the irrotational flow of a stream with the general velocity — U parallel to the axis, past a fixed spherical surface r=a, is given by +-lw(i-£) ( 15 > * This result is due to Stokes, I.e. f "On a Spherical Vortex," Phil. Trans. A, clxxxv. (1894). 246 Yortex Motion [chap, vii The two values of y\r agree when r=a ; this makes the normal velocity zero on both sides. In order that the tangential velocity may be continuous, the values of dyjf/dr must also agree. Remembering that or=r sin 0, this gives A = — § U/a 2 , and therefore a> = ^-U^ja 2 (16) The sum of the strengths of the vortex-filaments composing the spherical vortex is 5 Ua. The figure shews the stream-lines, both inside and outside the vortex ; they are drawn, as usual, for equidistant values of x//\ If we impress on everything a velocity U parallel to x, we get a spherical vortex advancing with constant velocity U through a liquid which is at rest at infinity. By the formulae of Art. 162, we readily find that the square of the ' mean-radius ' of the vortex is fa 2 , the 'impulse' 2irpa 3 U, and the energy is ^-Trpa^U 2 . As explained in Art. 146, it is quite unnecessary to calculate formulae for the pressure, in order to assure ourselves that this is continuous at the surface of the vortex. The con- tinuity of the pressure is already secured by the continuity of the velocity, and the constancy of the circulation in any moving circuit. 166. As already stated, the theory of vortex motion was originated by Helmholtz in 1858. It acquired additional interest when, in 1867, Kelvin suggested* the theory of vortex atoms. As a physical theory, this has long been abandoned, but it gave rise to a great number of interesting investi- gations, to which some reference should be made. We may mention the investigations as to the stability and the periods of vibration of rectilinear f and annular J vortices ; the similar investigations relating to hollow vortices (where the rotationally moving core is replaced by a vacuum§); and the cal- culations of the forms of boundary of a hollow vortex which are consistent with steady motion ||. A summary of some of the leading results has been given by Love IT. * I.e. ante p. 222. + Sir W. Thomson, I.e. ante p. 230. % J. J. Thomson, I.e. ante p. 216 ; Dyson, Phil. Trans. A, clxxxiv. 1041 (1893). § Sir W. Thomson, I.e. ; Hicks, "On the Steady Motion and the Small Vibrations of a Hollow Vortex," Phil. Trans. 1884; Pocklington, "The Complete System of the Periods of a Hollow Vortex King," Phil. Trans. A, clxxxvi. 603 (1895); Carslaw, "The Fluted Vibrations of a Circular Vortex-King with a Hollow Core," Proc. Lond. Math. Soc. (1) xxviii. 97 (1896). || Hicks, I.e.; Pocklington, "Hollow Straight Vortices," Camb. Proc. viii. 178 (1894). IF I.e. ante p. 192. 165-166 a] Bjerknes' Theorem 247 166 a. The dynamical theorems of the present chapter all depend on the constancy of the circulation in a moving circuit. It is postulated (Art. 146) that the extraneous forces if any are conservative, and also that the fluid is either homogeneous and incompressible, or subject to a definite relation between the pressure and the density. There are of course many natural conditions, especially in Meteorology, in which this latter assumption does not hold. If we proceed as in Art. 33 without making this assumption we find, for the rate of change of the circu- lation in a moving circuit, ^J{udx + vdy + wdz) = -j8\^dx + ^dy+^dzj t (1) where s (= 1/p) is the reciprocal of the density, or the 'bulkiness', of the fluid. The line-integral on the right hand may be converted into a surface-integral over any area bounded by the circuit, by Stokes' theorem ; thus jyl(urda> + vdy+wdz)= \\{IP + mQ + nlt)dS, (2) where *-|M C = fM fi-?4*4 (3) d(y,z) d(z t a>) o{x,y) Now consider the vector whose components are P, Q y R. It is solenoidal, in virtue of the relation dP dQ dB dx + d y + a* ' w and its direction is given by the intersections of the surfaces p = const., s = const. If we imagine a series of surfaces of equal pressure to be drawn for equal infinitesimal intervals 8p } and likewise a series of surfaces of equal bulkiness for equal infinitesimal intervals 8s, these will divide the field into a system of tubes whose cross-sections are infinitesimal parallelograms. It is easy to shew that if 8% is the area of one of these parallelograms *J(P*+Q 2 + R 2 )82 = 8p8s (5) Hence the product of the vector (P, Q, R,) into the cross-section is not only uniform along any tube, but is the same for all the tubes. The equation (2) then shews that the rate of change of the circulation round a moving circuit is proportional to the number of the aforesaid tubes which it embraces *. * V. Bjerknes, Vid.-Selsk. Skrifter, Kristiania, 1918. An independent proof is attributed to Silberstein (1896). Another theorem of a less simple character is given by Bjerknes, relating to the circulation of momentum jp (udx + vdy + wdz). Some applications of the theorems to meteorological and other phenomena are explained in Stockholm, Ah. Handl. xxxi. (1898). 248 Vortex Motion [chap, vii Clebsch's Transformation. 167. Another matter of some interest, which can however only be briefly touched upon, is Clebsch's transformation of the hydrodynamical equations*. It is easily seen that the component velocities at any one instant can be expressed in the forms — B+>£. — ! +x !> —£+4 t» where <£, X, fi are functions of x, y, z, provided the component rotations can be put in the forms dQ^jx) 3(X,/i) d(\ it i) t-d&zy *""a («,*)• i d{*,y) {) Now if the differential equations of the vortex-lines, viz. dx _dy _ dz . . 7~t~7' w be supposed integrated in the form a = const., /3 = const., , (4) where a, /3 are functions of x, y, z, we must have ^-^(y,*)' '-^fc*)* f "^a(*,y)' (& where P is some function of #, y, zf . Substituting these expressions in the identity d A+h + d A =0 dx + dy + dz ' we find W lJ ^=°> W which shews that P is of the form /(a, /3). If X, /x be any two functions of a, /3, we have 8.0,11) ' 3(X, M ) 3(«,g) 3(y >2 )-3(« ( » X 8(y )Z )' fflC -'' SC -' and the equations (5) will therefore reduce to the form (2), provided X, \i be chosen so that S8f$-/hA w which can obviously be satisfied in an infinity of ways. It is evident from (2) that the intersections of the surfaces X = const., fi= const, are the vortex-lines. This suggests that the functions X, \i which occur in (1) may be supposed to vary continuously with t in such a way that the surfaces in question move with the fluid % . Various analytical proofs of the possibility of this have been given ; the simplest, perhaps, is by means of the equations (2) of Art. 15, which give (as in Art. 17) udx + vdy + wdz=u da + v db + w dc- dx (8) It has been proved that we may assume, initially, u da+v db + w dc= —d<t>o + \d[x (9) Hence, considering space-variations at time t, we shall have udx-\-vdy + wdz= — c?0 + Xcfyi, (10) * "Ueber eine allgemeine Transformation d. hydrodynamischen Gleichungen," Crelle, liv. (1857) and lvi. (1859). See also Hill, Quart. Journ. Math. xvii. (1881), and Camb. Trans, xiv. (1883). f Cf. Forsyth, Differential Equations, Art. 174. % It must not be overlooked that on account of the insufficient determinacy of X, n these functions may vary continuously with t without relating always to the same particles of fluid, 167] HUTs Spherical Vortex 249 where (p=cp + x, and X, a have the same values as in (9), but are now expressed in terms of x, y, z, t. Since, in the ' Lagrangian ' method, the independent space- variables relate to the individual particles, this proves the theorem. On this understanding the equations of motion can be integrated, provided the extraneous forces have a potential, and that p is a function of p only. We have du o y . c* 9« , / 3X , 3X , d\\ da / 3u, da da\ d\ ~dx\ dt^ dt)* Dt dx Dt dx'' k ; and therefore, on the present assumption that D\jDt = 0, DajDt = 0, l * +Wt „.|-x| (U> by Art. 146 (5), (6). An arbitrary function of t is here supposed incorporated in d<p/dt. If the above condition be not imposed on X, a, we have, writing *-/4 + fc. + 0-j£ + xS (13) Dtdx Dtdx~ dx' Dt dy Dt dy~ dy ' Dt dz Dtdz~ dz ' '"^ ' Hence l ( ^H } =0, (15) d(x,y,z) shewing that H is of the form/(X, a, t) ; and Dt~ d*> Dt~d\ ""^ } * The author is informed that these equations were given in a Fellowship dissertation (Dublin) by Mr T. Stuart (1900). CHAPTER VIII TIDAL WAVES 168. One of the most interesting and successful applications of hydro- dynamical theory is to the small oscillations, under gravity, of a liquid having a free surface. In certain cases, which are somewhat special as regards the theory, but very important from a practical point of view, these oscillations may combine to form progressive waves travelling with (to a first approxi- mation) no change of form over the surface. The term 'tidal,' as applied to waves, has been used in various senses, but it seems most natural to confine it to gravitational oscillations possessing the characteristic feature of the oceanic tides produced by the action of the sun and moon. We have therefore ventured to place it at the head of this Chapter, as descriptive of waves in which the motion of the fluid is mainly horizontal, and therefore (as will appear) sensibly the same for all particles in a vertical line. This latter circumstance greatly simplifies the theory. It will be convenient to recapitulate, in the first place, some points in the general theory of small oscillations which will receive constant exemplification in the investigations which follow*. The theory has reference in the first instance to a system of finite freedom, but the results, when properly inter- preted, hold good without this restriction f. Let q ly q%, ... q n be n generalized co-ordinates serving to specify the con- figuration of a dynamical system, and let them be so chosen as to vanish in the configuration of equilibrium. The kinetic energy T will be a homogeneous quadratic function of the generalized velocities q lt q 2 , ... q nt say 2T=a U 5i 2 + a 2 2?2 2 +... + 2a 12 g 1 g 2 + ..., (1) where the coefficients are in general functions of the co-ordinates q 1} q 2 , ... q n , but may in the application to small motions be supposed constant, and to have the values corresponding to q 1} q 2 , ... q n — 0. Again, if (as we shall suppose) the system is 'conservative,' the potential energy Fof a small displacement is a homogeneous quadratic function of the component displacement q lt q 2} ... q n> with (on the same understanding) constant coefficients, say 2V = c u gi 2 4- c 22 q 2 2 + ... + 2ci2?ig 2 + (2) * For a fuller account of the general theory see Thomson and Tait, Arts. 337, ...; Kayleigh, Theory of Sound, c. iv.; Bouth, Elementary Rigid Dynamics (6th ed.), London, 1897, c. ix.; Whittaker, Analytical Dynamics, c. vii. ; Lamb, Higher Mechanics, 2nd ed., Cambridge, 1929. t The steps by which a rigorous transition can be made to the case of infinite freedom have been investigated by Hilbert, Gott. Nachr. 1904. 168] Free Oscillations 251 By a real* linear transformation of the co-ordinates q lf q 2 , ... q n it is possible to reduce T and V simultaneously to sums of squares; the new variables thus introduced are called the 'normal co-ordinates' of the system. In terms of these we have 2T=a 1 j 1 2 + a 2 ^+... + a„^ (3) 2F= c iqi 2 + c 2 q 2 * + ... + c n q n 2 ..(4) The coefficients a 1} a 2 , ... a n are called the 'principal coefficients of inertia'; they are necessarily positive. The coefficients Ci, c 2 , ... c n may be called the 'principal coefficients of stability'; they are all positive when the undisturbed configuration is stable. When given extraneous forces act on the system, the work done by these during an arbitrary infinitesimal displacement Aq 1} Aq 2 ,... Ag n may be ex- pressed in the form QiA^+QaA^+.-. + ^A^ (5) The coefficients Qi, Q 2 , .-.Q n are then called the 'normal components of disturbing force.' In the application to infinitely small motions Lagrange's equations d dT dT dV ~ r , /ax take the form dir'qi + a 2r q 2 + ... + c lr q-i -I- c 2r q2 + ... - Qr (7) or, in the case of normal co-ordinates, a r q r + c r q r =Q r (8) It is easily seen from this that the dynamical characteristics of the normal co-ordinates are (1°) that an impulse of any normal type produces an initial motion of that type only, and (2°) that a steady disturbing force of any type maintains a displacement of that type only. To obtain the free motions of the system we put Q r = 0. Solving (8), we find q r — A r cos (cr r t + e r ), (9) where o~r = {o r ja r )^, (10) and A r , € r are arbitrary constants f. Hence a mode of free motion is possible in which any normal co-ordinate q r varies alone, and the motion of any particle of the system, since it depends linearly on q ry will be simple-harmonic, of period 27r/o- r ; moreover the particles will keep step with one another, passing simultaneously through their equilibrium positions. The several modes of this character are called the 'normal modes' of vibration of the system; their * The algebraic proof of this involves the assumption that one at least of the functions T, V is essentially positive. In the present case T of course fulfils this condition. t The ratio c\1-ir measures the 'frequency' of the oscillation. It is convenient to have a name for the quantity a itself; the term 'speed' has been used in this sense by Kelvin and G. H. Darwin in their researches on the Tides. 252 Tidal Waves [chap, vm number is equal to that of the degrees of freedom, and any free motion what- ever of the system may be obtained from them by superposition, with a proper choice of the 'amplitudes' (A r ) and 'epochs' (e r ). It is seen from (10) that in any normal mode the mean values (with respect to time) of the kinetic and potential energies are equal. In certain cases, viz. when two or more of the free periods (27r/a) of the system are equal, the normal co-ordinates are to a certain extent indeterminate, i.e. they can be chosen in an infinite number of ways. By compounding the corresponding modes, with arbitrary amplitudes and epochs, we obtain a small oscillation in which the motion of each particle is the resultant of simple- harmonic vibrations in different directions, and is therefore, in general, elliptic- harmonic, with the same period. This is exemplified in the spherical pendulum ; an important instance in our own subject is that of progressive waves in deep water (Chapter IX.). If any of the coefficients of stability (c r ) be negative, the value of a r is a pure imaginary. The circular function in (9) is then replaced by real exponentials, and an arbitrary displacement will in general increase until the assumptions on which the approximate equation (8) is based becomes untenable. The undisturbed configuration is then reckoned as unstable. The necessary and sufficient condition of stability (in the present sense) is that the potential energy V should be a minimum in the configuration of equilibrium. To find the effect of disturbing forces, it is sufficient to consider the case where Q r varies as a simple-harmonic function of the time, say Q r = C r cos (at + e), (11) where the value of a is now prescribed. Not only is this the most interesting case in itself, but we know from Fourier's Theorem that, whatever the law of variation of Q r with the time, it can be expressed by a series of terms such as (11) A particular integral of (8) is then &"* ^7 cos(cr* + e) (12) This represents the 'forced oscillation' due to the periodic force Q r . In it the motion of every particle is simple-harmonic, of the prescribed period 27r/a, and the extreme displacements coincide in time with the maxima and minima of the force. A constant force equal to the instantaneous value of the actual force (11) would maintain a displacement Q q r = -1 cos (at + e), (13) c r the same, of course, as if the inertia-coefficient a r were null. Hence (12) may be written q ^T^}^ (14) 168] Forced Oscillations 253 where g t has the value (10). This very useful formula enables us to write down the effect of a periodic force when we know that of a steady force of the same type. It is to be noticed that q r and Q r have the same or opposite phases according as cr $ cr r , that is, according as the period of the disturbing force is greater or less than the free period. A simple example of this is furnished by a simple pendulum acted on by a periodic horizontal force. Other important illustrations will present themselves in the theory of the tides *. When g is very great in comparison with a r , the formula (12^ becomes ?r = -^cos(<j£ + e); (15) the displacement is now always in the opposite phase to the force, and depends only on the inertia of the system. If the period of the impressed force be nearly equal to that of the normal mode of order r, the amplitude of the forced oscillation, as given by (14), is very great compared with q r . In the case of exact equality, the solution (12) fails, and must be replaced by C t gv = 2^- sin (o-£ + e) (16) This gives an oscillation of continually increasing amplitude, and can therefore only be accepted as a representation of the initial stages of the disturbance. Another very important property of the normal modes may be noticed. If by the introduction of frictionless constraints the system be compelled to oscillate in any other prescribed manner, the configuration at any instant can be specified by one variable, which we will denote by 6. In terms of this we shall have q r =B r 6, where the quantities B r are certain constants. This makes 2T=(B 1 *a 1 +B 2 *a 2 +...+B n *a n )0*, (17) 2F=(^ Cl+ ^ C2 + ...+42 c ^2 (18) If a cos (o-^ + e), the constancy of the energy (T+ V) requires i _ B 1 *c 1 + Bfc a +...+B n *c n . . °" B 1 *a 1 + B 2 *a 2 + ...+B 1 ?a n ^ Hence o- 2 is intermediate in value between the greatest and least, of the quantities cr/a r ; in other words, the frequency of the constrained oscillation is intermediate between the greatest and least frequencies corresponding to the normal modes of the system. In par- ticular, when a system is modified by the introduction of a constraint, the frequency of the slowest natural oscillation is increased. Moreover, if the constrained type differ but slightly from a normal type (r), o- 2 will differ from c r /a r by a small quantity of the second order. This gives a method of estimating approximately the frequency in cases where the normal types cannot be accurately determined f. Examples will be found in Arts. 191, 259. * Cf. T. Young, "A Theory of Tides," Nicholson's Journal, xxxv. (1813) [Miscellaneous Works, London, 1854, ii. 262]. t Eayleigh, "Some General Theorems relating to Vibrations," Proc. Lond. Math. Soc. iv. 357 (1874) [Papers, i. 170], and Theory of Sound, c. iv. The method was elaborated by Eitz, Journ. fur Math, exxxv. 1 (1908), and Ann. der Physik, xxviii. (1909) [Gesammelte Werke, Paris, 1911, pp. 192, 265]. 254 Tidal Waves [chap, fiii It may further be shewn that in the case of a partial constraint, which merely reduces the degree of freedom from n to n — 1, the periods of the modified system separate those of the original one* It had been already remarked by Lagrange f that if in the equations of type (7), where the co-ordinates are not assumed to be normal, we put Q r =0, and assume q r = A r cos (o-t + e), (20) the resulting equations are identical with those which determine the stationary values of the expression 9 ^ c n AS+c 22 A 2 *+ ... +gci^M2+ ... V(A,-A) . v a a n A l *+a 22 A 2 *+...+2a l2 A l A 2 + ..: T (A, A)' l ; say. Since T(A, A) is essentially positive the denominator cannot vanish, and the expression has therefore a minimum value. It is moreover possible, starting from this property, to construct a proof that the n values of o- 2 are all real % . They are obviously all positive if V be essentially positive. Rayleigh's theorem is also closely related to the Hamiltonian formula (3) of Art. 135, as we may see by assuming q r =A r sin at, (22) and taking t =0, ^— 2jt/ot. Cf. Art. 205 a. The modifications which are introduced into the theory of small oscillations by the consideration of viscous forces will be noticed in Chapter XL Long Waves in Canals. 169. Proceeding now to the special problem of this chapter, let us begin with the case of waves travelling along a straight canal, with horizontal bed, and parallel vertical sides. Let the axis of x be parallel to the length of the canal, that of y vertical and upwards, and let us suppose that the motion takes place in these two dimensions x, y. Let the ordinate of the free surface, corresponding to the abscissa x, at time t, be denoted by y + tj, where y is the ordinate in the undisturbed stalfe. As already indicated, we shall assume in all the investigations of this Chapter that the vertical acceleration of the fluid particles may be neglected, or, more precisely, that the pressure at any point (x, y) is sensibly equal to the statical pressure due to the depth below the free surface, viz. p-po = gp(yo + y-y), (i) where p is the (uniform) external pressure. S-»S < 2 > This is independent of y, so that the horizontal acceleration is the same for all particles in a plane perpendicular to x. It follows that all particles which once lie in such a plane always do so ; in other words, the horizontal velocity u is a function of x and t only. **Kouth, Elementary Rigid Dynamics, Art. 67; Rayleigh, Theory of Sound (2nd ed.), Art. 92 a; Whittaker, Analytical Dynamics, Art. 81. t Mecanique Analytique (Bertrand's ed.), i. 331; Oeuvres, xi. 380. % See Poincare, Journ. de Math. (5), ii. 83 (1896); Lamb, Higher Mechanics, 2nd ed., Art. 92. 168-169] Waves in Uniform Canal 255 The equation of horizontal motion, viz. du du _ Idp dt dx pdx' is further simplified in the ca3e of infinitely small motions by the omission of the term udu/dx, which is of the second order, so that S"—'S (3) Now let S^fudt; i.e. £ is the time-integral of the displacement past the plane x, up to the time t. In the case of small motions this will, to the first order of small quantities, be equal to the displacement of the particles which originally occupied that plane, or again to that of the particles which actually occupy it at time t The equation (3) may now be written *f— ,*S (4) The equation of continuity may be found by calculating the volume of fluid which has, up to time t, entered the space bounded by the planes x and x + Bx; thus, if h be the depth and b the breadth of the canal, — a~ (£hb) 8® — rjbSx, *— *i (5) The same result comes from the ordinary form of the equation of continuity, viz. te + ty = ° (6) •(7) __ fvdu , du Thus '--./.B*— »» if the origin be (for the moment) taken in the bottom of the canal. This formula is of interest as shewing, as a consequence of our primary assumption, that the vertical velocity of any particle is simply* proportional to its height above the bottom. At the free surface we have y = h + rj, v = drj/dt, whence (neglecting a product of small quantities) di-- h fadl (8) From this (5) follows by integration with respect to t. Eliminating tj between (4) and (5), we obtain dt*~ g dx* The elimination of f gives an equation of the same form, viz. w*= gh w The above investigation can readily be extended to the case of a uniform 3-*s » sf'-"3 <>») 256 Tidal Waves [chap, vin canal of any form of section *. If the sectional area of the undisturbed fluid be S, and the breadth at the free surface b, the equation of continuity is -^S)8x = v bhx, (11) whence rj = — h^, (12) as before, provided h — S/b, i.e. h now denotes the mean depth of the canal. The dynamical equation (4) is of course unaltered. 170. The equation (9) is of a well-known type which occurs in several physical problems, e.g. the transverse vibrations of strings, and the motion of sound-waves in one dimension. To integrate it, let us write, for shortness, «-#) fltt and x — ct = x 1} x-\-ct — x^. In terms of x± and x z as independent variables, the equation takes the form The complete solution is therefore £ = F(x-ct)+f(x + ct), (14) where F, f are arbitrary functions. The corresponding values of the particle- velocity and of the surface-elevation are given by \ ^-F'{x-ct)-f'{x + ct). The interpretation of these results is simple. Take first the motion repre- sented by the first term in (14), alone. Since F (x — ct) is unaltered when t and x are increased by t and ct, respectively, it is plain that the disturbance which existed at the point x at time t has been transferred at time t + t to the point x + ct. Hence the disturbance advances unchanged with a constant velocity c in space. In other words we have a 'progressive wave' travelling with velocity c in the direction of ^-positive. In the same way the second term of (14) represents a progressive wave travelling with velocity c in the direction of ^-negative. And it appears, since (14) is the complete solution of (9), that any motion whatever of the fluid, which is subject to the conditions laid down in the preceding Art., may be regarded as made up of waves of these two kinds. * Kelland, Trans. B. S. Edin. xiv. (1839). c .(15) 169-171] Initial Conditions 257 The velocity (c) of propagation is, by (13), that 'due to' half the depth of the undisturbed fluid*. The following table giving, in round numbers and assuming <7=32f/s, the velocity of wave-propagation for various depths, will be of interest later in connection with the theory of the tides. The last column gives the time a wave would take to travel over a distance equal to the earth's circumference (%ira). In order that a 'long' wave should traverse this distance in 24 hours, the depth would have to be about 14 miles. It must be borne in mind that these numerical results are only applicable to waves satisfying the conditions above postulated. The meaning of these conditions will be examined more particularly in Art. 172. h c c 2ira/c (feet) (feet per sec.) (sea-miles per hour) (hours) 3121 100 60 360 1250 200 120 180 5000 400 240 90 11250f 600 360 60 20000 800 480 45 171. To trace the effect of an arbitrary initial disturbance, let us suppose that when t = we have \=$(oo), j-f(*j (16) The functions F', f which occur in (15) are then given by *"(•) — *{*(•) + + (•)},} Q7 x /(*>- *{*(»)-*(•)}.; v ' Hence if we draw the curves y — rji, y = 7] 2 , where i?i = iM* (*) + *(*)}») (1S) % = iM*(aO -*(*)},} the form of the wave-profile at any subsequent instant t is found by displacing these curves parallel to x, through spaces ± ct, respectively, and adding (alge- braically) the ordinates. If, for example, the original disturbance is confined to a length I of the axis of oo, then after a time l/2c it will have broken up into two progressive waves of length I, travelling in opposite directions. In the particular case where in the initial state f = 0, and therefore cj> (a?) = 0, we have t) X = yz] the elevation in each of the derived waves is then exactly half what it was, at corresponding points, in the original disturbance. It appears from (16) and (17) that if the initial disturbance be such that ? = ± v/h • c > the motion will consist of a wave system travelling in one direction only, since one or other of the functions F' and f is then zero. * Lagrange, Nouv. mem. de V Acad, de Berlin, 1781 [Oeuvres, i. 747]. t This is probably comparable in order of magnitude with the mean depth of the ocean. 258 Tidal Waves [chap, vin It is easy to trace the motion of a surface-particle as a progressive wave of either kind passes it. Suppose, for example, that %=*F(w-ct), (19) and therefore %~ c h (^ The particle is at rest until it is reached by the wave ; it then moves forward with a velocity proportional at each instant to the elevation above the mean level, the velocity being in fact less than the wave-velocity c, in the ratio of the surface-elevation to the depth of the water. The total displacement at any time is given by Z=l\ v cdt. p Di := -dy- 9p ' This integral measures the volume, per unit breadth of the canal, of the portion of the wave which has up to the instant in question passed the particle. Finally, when the wave has passed away, the particle is left at rest in advance of its original position at a distance equal to the total volume of the elevated water divided by the- sectional area of the canal. 172. We can now examine under what circumstances the solution expressed by (14) will be consistent with the assumptions made provisionally in Art. 169. The exact equation of vertical motion, viz. Dv _ dp gives, on integration with respect to y, p-po=gp(yo + v-y)-p) jy t dy (21) This may be replaced by the approximate equation (1), provided fih be small compared with gi), where /3 denotes the maximum vertical acceleration. Now in a progressive wave, if X denote the distance between two consecutive nodes {i.e. points at which the wave-profile meets the undisturbed level), the time which the corresponding portion of the wave takes to pass a particle is X/c, and therefore, provided the gradient dr)/d% is everywhere small, the vertical velocity will be of the order rjc/X*, and the vertical acceleration of the order ?7C 2 /X 2 , where j] is the maximum elevation (or depression). Hence /3h will be small compared with grj, provided h 2 /\ 2 is a small quantity. Waves whose slope is gradual, and whose length X is large compared with the depth h of the fluid, are called 'long' waves. Again, the restriction to infinitely small motions, made in equation (3), consisted in neglecting udu/dx in comparison with du/dt. In a progressive * Hence, comparing with (20), we see that the ratio of the maximum vertical to the maximum horizontal velocity is of the order hj\. 171-173] Airy's Method 259 wave we have du/dt = + cdu/dx; so that u must be small compared with c, and therefore, by (20), r) must be small compared with h. It is to be observed that this condition is altogether distinct from the former one, which may be legitimate in cases where the motion cannot be regarded as infinitely small. See Art. 187. The preceding conditions will of course be satisfied in the general case represented by equation (14), provided they are satisfied for each of the two progressive waves into which the disturbance can be analysed. 173. There is another, although on the whole a less convenient, method of investigating the motion of 'long' waves, in which the Lagrangian plan is adopted of making the co-ordinates refer to the individual particles of the fluid. For simplicity, we will consider only the case of a canal of rectangular section*. The fundamental assumption that the vertical acceleration may be neglected implies as before that the horizontal motion of all particles in a plane perpendicular to the length of the canal will be the same. We therefore denote by x + f the abscissa at time t of the plane of particles whose undisturbed abscissa is x. If rj denote the elevation of the free surface, in this plane, the equation of motion of unit breadth of a stratum whose thickness (in the un- disturbed state) is Bx will be where the factor (dp/dx) . Bx represents the pressure-difference for any two opposite particles x and x + Bx on the two faces of the stratum, while the factor h + rj represents the area of the stratum. Since we assume that the pressure about any particle depends only on its depth below the free surface we may write dp _ drj so that our dynamical equation is 3--»(>+-9& » The equation of continuity is obtained by equating the volumes of a stratum, consisting of the same particles, in the disturbed and undisturbed conditions respectively, viz. Bx + ^- Bx ) (h + v) = hBx, (■♦a: < 2 > * Airy, Encyc. Metrop. "Tides and Waves," Art. 192 (1845); see also Stokes, "On Waves," Camb. and Dub. Math. Journ. iv. 219 (1849) [Papers, ii. 222]. The case of a canal with sloping sides has been treated by McCowan, "On the Theory of Long Waves...," Phil. Mag. (5), xxxv. 250(1892). 260 Tidal Waves [chap, viii Between equations (1) and (2) we may eliminate either rj or f ; the result in terms of f is the simpler, being ay- 8 2 f , da? dt 2 gh-, ^r 3 (3) ('♦B This is the general equation of 'long' waves in a uniform canal with vertical sides *. So far the only assumption in the present investigation is that the vertical acceleration of the particles may be neglected in calculating the pressure. If we now assume, in addition, that rj/h is a small quantity, the equations (2) and (3) reduce to ?— *S« w and S=^S < 5) The elevation rj now satisfies an equation of the same form, viz. w =gh w (6) These are in conformity with our previous results; for the smallness of 5f /3a? means that the relative displacement of any two particles is never more than a minute fraction of the distance between them, so that (to a first ap- proximation) it is now immaterial whether the variable x be supposed to refer to a plane fixed in space, or to one moving with the fluid. 174. The potential energy of a wave, or system of waves, due to the elevation or depression of the fluid above or below the mean level is, per unit breadth, gpjfydxdy, where the integration with respect to y is to be taken between the limits and rj, and that with respect to x over the whole length of the waves. Effecting the former integration, we get igpfv 2 dx (1) The kinetic energy is \ph^dx (2) In a system of waves travelling in one direction only we have so that the expressions (1) and (2) are equal; or the total energy is half potential, and half kinetic. This result may be obtained in a more general manner, as follows js Any progressive wave may be conceived as having been originated by the splitting * Airy, I.e. t Bayleigh, "On Waves," Phil. Mag. (5), i. 257 (1876) [Papers, i. 251]. 173-175] Energy of Long Waves 261 up, into two waves travelling in opposite directions, of an initial disturbance in which the particle-velocity was everywhere zero, and the energy therefore wholly potential. It appears from Art. 171 that the two derived waves are symmetrical in every respect, so that each must contain half the original store of energy. Since, however, the elevation at corresponding points is for each derived wave exactly half that of the original disturbance, the potential energy of each will by (1) be one-fourth of the original store. The remaining (kinetic) part of the energy of each derived wave must therefore also be one-fourth of the original quantity. 175. If in any case of waves travelling in one direction only, without change of form, we impress on the whole mass a velocity equal and opposite to that of propagation, the motion becomes steady, whilst the forces acting on any particle remain the same as before. With the help of this artifice, the laws of wave-propagation can be investigated with great ease *. Thus, in the present case we shall have, by Art. 22 (5), at the free surface, ^ = const. -#(/* + 77) -l^ 2 , (1) where q is the velocity. If the slope of the wave-profile be everywhere gradual, and the depth h small compared with the length of a wave, the horizontal velocity may be taken to be uniform throughout the depth, and approximately equal to q. Hence the equation of continuity is q(h + v ) = ch, (2) c being the velocity, in the steady motion, at places where the depth of the stream is uniform and equal to h. Substituting for q in (1), we have V £ = const. -^(l + Q-ic^l + D (3) Hence if rj/h be small, the condition for a free surface, viz. p = const., is satisfied approximately, provided c 2 = gh, (4) which agrees with our former result. The present method also accounts very simply for the relation between particle-velocity and surface-elevation already found in Art. 171. From (2) we have, approximately, r'^-S < 5 > Hence in the wave-motion the particle-velocity relative to the undisturbed water is crj/h in the direction of propagation. When the elevation rj, though small compared with the wave-length, is not * Eayleigh, I.e. 262 Tidal Waves [chap, viii regarded as infinitely small, a closer approximation to the wave- velocity is secured if in (4) we replace h by ij + h. This gives a wave- velocity approximately, where c = \/(gh)> relative to the fluid in the immediate neigh- bourhood. Since this fluid has itself a velocity c r]/h, the velocity of propagation in space is approximately <*4l)> w a result due substantially to Airy*. It follows that a wave of the type now under consideration cannot be propagated entirely without change of profile, since the speed varies with the height. Another proof of (6) will be given presently when we come to consider specially the theory of waves of finite amplitude (Art. 187). 176. It appears from the linearity of the approximate equations that, in the case of sufficiently low waves, any number of independent solutions may be superposed. For example, having given a wave of any form travelling in one direction, if we superpose its image in the plane x = 0, travelling in the opposite direction, it is obvious that in the resulting motion the horizontal velocity will vanish at the origin, and the circumstances are therefore the same as if there were a fixed barrier at this point. We can thus understand the reflection of a wave at a barrier; the elevations and depressions are reflected unchanged, whilst the horizontal velocity is reversed. The same results follow from the formula %=F(ct-x)-F(ct + x), (1) which is evidently the most general value of f subject to the condition that £=0 for # = 0. We can further investigate without much difficulty the partial reflection of a wave at a point where there is an abrupt change in the section of the canal. Taking the origin at the point in question, we may write, for the negative side, *-*KM'+3. -^KH'K) (2) and for the positive side *-♦(«-£), «.-« ♦(<-£), (3) where the function F represents the original wave, and /, cf> the reflected and transmitted portions respectively. The constancy of mass requires that at the point x=Q we should have b-Ji x U\ = b 2 h 2 u 2 , where b u b 2 are the breadths at the surface, and A 1? h 2 are the mean depths. We must also have at the same point ^ — r]^ on account of the continuity of pressure t. These conditions give b ^{F(t)-f(t)}J-^cf>(t\ F(t)+f(t)=4>(t\ c x c 2 * "Tides and Waves," Art. 208. f It will be understood that the problem admits only of an approximate treatment, on account of the rapid change in the character of the motion near the point of discontinuity. The nature 175-177] Forced Waves 263 We thence find that the ratios of the elevations in corresponding parts of the reflected and incident waves, and of the transmitted and incident waves, are f ^ b 1 c 1 -b 2 c 2 <f>__ 26^! F &1C1+&2V P b 1 ci + b 2 c 2 1 respectively. The reader may easily verify that the energy contained in the reflected and transmitted waves is equal to that of the original incident wave. 177. Our investigations, so far, relate to cases of free waves. When, in addition to gravity, small disturbing forces X, Y act on the fluid, the equation of motion is obtained as follows. We assume that within distances comparable with the depth h these forces vary only by a small fraction of their total value. On this understanding we have, in place of Art. 169 (1), P-=-£° = (g-Y)(y n + v -y), (1) and theretore i| = (gr - Y) g - (y, + r, - y) g". We assume that Y is small compared with g, and (for the reason just stated) that hd Yfbx is small compared with X. Hence, with sufficient approximation, the equation of horizontal motion , viz. 9~it*' » reduces to the form dt*— g dx + X > ' (3) where, moreover, X may be regarded as a function of x ana t only. The equation of continuity is the same as in Art. 169, viz. Hence, on elimination of rj, *--*8 <*> J-ghJ + x (5) The horizontal component of the disturbing force is alone important. If the disturbing influence consists of a variable surface -pressure (p ), the equation (3) is replaced by 8t 2 9 dx p dx' K ' of the approximation implied in the above assumptions will become more evident if we suppose the suffixes to refer to two sections S 1 and S 2 , one on each side of the origin 0, at distances from which, though very small compared with the wave-length, are yet moderate multiples of the transverse dimensions of the canal. The motion of the fluid will be sensibly uniform over each of these sections, and parallel to the length. The condition in the text then expresses that there is no sensible change of level between S 1 and S 2 . 264 Tidal Waves [chap, vm whilst (4) is unaltered. In the case of a travelling pressure, say p j=f(Ut-x\ (7) we find h-p(U*-gh) w The surface depression is in the same phase with the pressure, or the opposite, according as U > *J(gh). On the other hand, when it is the bottom which is disturbed, we have X = in (2), whilst the equation of continuity becomes "-"»=->>!' w where r) is the elevation of the bottom above the mean level. Thus in the case of a seismic wave Vo =f(Ut-x), (10) we find 178. The oscillations of water in a canal of uniform section, closed at both ends, may, as in the corresponding problem of Acoustics, be obtained by super- position of progressive waves travelling in opposite directions. It is more instructive, however, with a view to subsequent more difficult investigations, to treat the problem as an example of the general theory sketched in Art. 168. We have to determine f so as to satisfy W~ (? d^ + X ' (1) together with the terminal conditions that f = for x = and x = I, say. To find the free oscillations we put X — 0, and assume that f oc cos {at + e), where a is to be found. On substitution we obtain g+S*-* w whence, omitting the time-factor, „ . . ax , D ax t = A sin V B cos — . * c c The terminal conditions give B=0, and al/c = rir, (3) where r is integral. Hence the normal mode of order r is given by f =^4 r sin -^-cos f— j- +e r J, (4) where the amplitude A r and epoch e r are arbitrary. 177-179] Waves in a Finite Canal 265 In the slowest oscillation (r = 1), the water sways to and fro, heaping itself up alternately at the two ends, and there is a node at the middle (x = \l). The period (21 /c) is equal to the time a progressive wave would take to traverse twice the length of the canal. The periods of the higher modes are respectively J, J, J, ... of this, but it must be remembered, in this and in other similar problems, that our theory ceases to be applicable when the length Ijr of a semi-undulation becomes comparable with the depth h. On comparison with the general theory of Art. 168, it appears that the normal co-ordinates of the present system are quantities q lt q 2} ... q n such that when the system is displaced according to any one of them, say q r , we have f = q r sin -j- ; and we infer that the most general displacement of which the system is capable (subject to the conditions presupposed) is given by y v . V7TX , f = %.sin-£- , (5) where q lt q 2 , ... q n are arbitrary. This is in accordance with Fourier's Theorem. When expressed in terms of the normal velocities and the normal co-ordi- nates, the expressions for T and V must reduce to sums of squares. This is easily verified, in the present case, from the formula (5). Thus if S denote the sectional area of the canal, we find 2T=o8 f ?dx = 2,a r q 2 , 2V = go f [ r ) 2 dx = $c r q 2 , (6) Jo hJo where a r = ^pSl, c r = %r 2 7r 2 gphS/l (7) It is to be noted that, on the present reckoning, the coefficients of stability (cv) increase with the depth. Conversely, if we assume from Fourier's Theorem that (5) is a sufficiently general expression for the value of f at any instant, the calculation just indicated shews that the coefficients q r are the normal co-ordinates; and the frequencies can then be found from the general formula (10) of Art. 168; viz. we have <rr = (Cr/ar)* = rTr(gh)t/l, (8) in agreement with (3). 179. As an example of forced waves we take the case of a uniform hori- zontal force X=/cos(o-£ + e) (9) This will illustrate, to a certain extent, the generation of tides in a land- locked sea of small dimensions. 266 Tidal Waves [chap, viii Assuming that f varies as cos {at 4- e), and omitting the time-factor, the equation (1) becomes the solution of which is 9^ c 2 * c 2 ' f = -4 + 2)sin— + #008 — (10) a 2 c c The terminal conditions give E=l, 2>si^ = (l-co^)/ (11) Hence, unless sin al/c = 0, we have D =fja 2 . tan al/2c, so that .(12) f. 2/ . ax . a- (I — x) , . f - ? cos 4^/o) Sm 23 Sm "V- • ° 0S ^ + e) ' j hf . a{oc — hl) , , and 7) = £= — — - sm — — . cos (at + e). crc cos (^ aijC) c If the period of the disturbing force be large compared with that of the slowest free mode, al/2c will be small, and the formula for the elevation becomes 7i = *-(x-kl)coa(<rt+e) t (13) approximately, exactly as if the water were devoid of inertia. The horizontal displacement of the water is always in the same phase with the force, so long as the period is greater than that of the slowest free mode, or crl/c < it. If the period be diminished until it is less than the above value, the phase is reversed. When the period is exactly equal to that of a free mode of odd order (r = 1, 3, 5, ...), the above expressions for f and tj become infinite, and the solution fails. As pointed out in Art. 168, the interpretation of this is that, in the absence of dissipative forces, the amplitude of the motion becomes so great that our fundamental approximations are no longer justified. If, on the other hand, the period coincide with that of a free mode of even order (r= 2, 4, 6, ...), we have sin al/c = 0, cos al/c — 1, and the terminal conditions are satisfied independently of the value of D. The forced motion may then be represented by* f = _^ s in 2 ^cos(<7*+e) (14) This example illustrates the fact that the effect of a disturbing force may sometimes be conveniently calculated without resolving the force into its 'normal components.' * In the language of the general theory, the impressed force has here no component of the particular type with which it synchronizes, so that a vibration of this type is not excited at all. In the same way a periodic pressure applied at any point of a stretched string will not excite any fundamental mode which has a node there» even though it synchronize with it. 179-180] Canal Theory of the Tides 267 Another very simple case of forced oscillations, of some interest in connection with tidal theory, is that of a canal closed at one end and communicating at the other with an open sea in which a periodic oscillation 7j = a cos (at + e) (15) is maintained. If the origin be taken at the closed end, the solution is obviously COS (ax/c) / . , \ /-./jv y = a / 7/ v . cos (o-S + e), (16) cos (trl/c) v ' x I denoting the length. If al/c be small the tide has sensibly the same amplitude at all points of the canal. For particular values of I (determined by cos crl/c = 0) the solution fails through the amplitude becoming infinite. Canal Theory of the Tides. 180. The theory of forced oscillations in canals, or on open sheets of water, owes most of its interest to its bearing on the phenomena of the tides. The 'canal theory,' in particular, has been treated very fully by Airy*. We will consider a few of the more interesting problems. The calculation of the disturbing effect of a distant body on the waters of the ocean is placed for convenience in an Appendix at the end of this Chapter. It appears that the disturbing effect of the moon, for example, at a point P of the earth's surface, may be represented by a potential XI whose approximate value is = i 3 ^?(i-co#^X (1) where M denotes the mass of the moon, D its distance from the earth's centre, a the earth's radius, y the 'constant of gravitation,' and SV the moon's zenith distance at the place P. This gives a horizontal acceleration d£l/ad^, or /sin 2*. (2) towards the point of the earth's surface which is vertically beneath the moon, where / = f^ (3) If E be the earth's mass, we may write g = yE/a 2 } whence /-? E (sl g -2'E'\D / Putting M/E=^ tf a/D = ^, this gives f/g = 8'57 x 10" 8 . When the sun is the disturbing body, the corresponding result is/Jg — S'7S x 10 -8 . It is convenient, for some purposes, to introduce a linear magnitude H, defined by H = af/g (4) * Encycl. Metrop. "Tides and Waves," Section vi. (1845). Several of the leading features of the theory had been made out, by very simple methods, by Young, in 1813 and 1823 [Works, ii. 262, 291]. 268 Tidal Waves [chap, viii If we put a = 21 x 10 6 feet, this gives, for the lunar tide, H= 1*80 ft., and for the solar tide H= > 79 ft. It is shewn in the Appendix that H measures the maximum range of the tide, from high water to low water, on the ' equilibrium theory/ 181. Take now the case of a uniform canal coincident with the earth's equator, and let us suppose for simplicity that the moon describes a circular orbit in the same plane. Let f be the displacement, relative to the earth's surface, of a particle of water whose mean position is in longitude (f>, measured eastwards from some fixed meridian. If co be the angular velocity of the earth's rotation, the actual displacement of the particle at time t will be f + acot, so that the tangential acceleration will be d 2 £/dt. If we suppose the 'centrifugal force' to be as usual allowed for in the value of g, the processes of Arts. 169, 177 will apply without further alteration. If n denote the angular velocity of the moon westward, relative to the fixed meridian*, we may write in Art. 180 (2) ^ = nt + <f> + e, so that the equation of motion is P= c2 j^ sin2 ^ + * +e > <*> The free oscillations are determined by the consideration that f is necessarily a periodic function of (£, its value recurring whenever <£ increases by 27r. It may therefore be expressed, by Fourier's Theorem, in the form f = 2 (P r cosr(f> + Q r smrcf>) (2) o Substituting in (1), with the last term omitted, it is found that P r and Q r must satisfy the equation J2D r 2 r 2 T*?- (3) The motion, in any normal mode, is therefore simple-harmonic, of period 2irajrc. For the forced waves, or tides, we find f = -1; nw si°2 (nt + $ + e), (4) c 2 H whence rj — J-j $—, - % cos 2 (nt + <£ + e) (5) c — 7i a The tide is therefore semi-diurnal (the lunar day being of course understood), and is 'direct' or 'inverted,' i.e. there is high or low water beneath the moon, according as c ^ na, in other words according as the velocity, relative to the * That is, n = w - n lf if n x be the angular velocity of the moon in her orbit. I8O-182] Equatorial Canal 269 earth's surface, of a point which moves so as to be always vertically beneath the moon, is less or greater than that of a free wave. In the actual case of the earth we have c 2 g h h ~2 — 2 ~ 2 * ~" — Oil—, rfar rra a a so that unless the depth of the canal were to greatly exceed such depths as actually occur in the ocean, the tides would be inverted. This result, which is sometimes felt as a paradox, comes under a general principle referred to in Art. 168. It is a consequence of the comparative slowness of the free oscillations in an equatorial canal of moderate depth. It appears from the rough numerical table on p. 257 that with a depth of 11250 feet a free wave would take about 30 hours to describe the earth's semi-circumference, whereas the period of the tidal disturbing force is only a little over 12 hours. The formula (5) is, in fact, a particular case of Art. 168 (14), for it may be written " = r^v^ (6) where rj is the elevation given by the 'equilibrium theory,' viz. 7}= %Hcos2 (nt + <f> + e), (7) and o- = 2n, <r = 2c/a. For such moderate depths as 10000 feet and under, n 2 a 2 is large compared with gh\ the amplitude of the horizontal motion, as given by (4), is then //4n 2 or g/4>n 2 a.H, nearly, being approximately independent of the depth. In the case of the lunar tide this amplitude is about 140 feet. The maximum elevation is obtained by multiplying by 2h/a; this gives, for a depth of 10000 feet, a height of only *133 of a foot. For greater depths the tides would be higher, but still inverted, until we reach the critical depth n 2 a 2 /g, which is about 13 miles. For depths beyond this limit, the tides become direct, and approximate more and more to the value given by the equilibrium theory *. 182. The case of a circular canal parallel to the equator can be worked out in a similar manner. If the moon's orbit be still supposed to lie in the plane of the equator, we find by spherical trigonometry cos Sr as sin 6 cos (nt + ft-f e), (1) where 6 is the co-latitude, and <j> the longitude. The disturbing force in longitude is therefore ^-5^ = -/sm0sin2(n£ + <£+e) (2) c 2 E sin 2 6 This leads to V = \ # _ n 2 a z sin 2 q cqs 2 (m* + ft + e) (3) * Cf. Young, I.e. ante p. 253. 270 Tidal Waves [chap, viii Hence if na > c the tide will be direct or inverted according as sin 6 ^ c/na. If the depth be so great that c> na y the tides will be direct for all values of 6. If the moon be not in the plane of the equator, but have a co-declination A, the formula (1) is replaced by- cos S- = cos 6 cos A + sin 6 sin A cos a, (4) where a is the hour-angle of the moon from the meridian of P. For simplicity, we will neglect the moon's motion in declination in comparison with the earth's angular velocity of rotation ; thus we put a = nt + <p + e, and treat A as constant. The resulting expression for the disturbing force along the parallel is found to be = ^/i/ — - /cos, # sin 2 A sin (nt + 6 + e) a sm dd<f) J v t * -/sin0sin 2 Asin2(?tf + + e) (5) We thence obtain c 2 H 71 = * c 2 -tt 2 a 2 sin 2 6> sin 26>sin 2A C0S (n * + * + e) ^ ^JL^ sin2ism2Acos2( ^^ +€) (6) The first term gives a 'diurnal' tide of period 2ir/n; this vanishes ana changes sign when the moon crosses the equator, i.e. twice a month. The second term represents a semi-diurnal tide of period tt/u, whose amplitude is now less than before in the ratio of sin 2 A to 1. 183. In the case of a canal coincident with a meridian we should have to take account of the fact that the undisturbed figure of the free surface is one of relative equilibrium under gravity and centrifugal force, and is therefore not exactly circular. We shall have occasion later on to treat the question of displacements relative to a rotating globe somewhat carefully; for the present we will assume by anticipation that in a narrow canal the disturbances are sensibly the same as if the earth were at rest, and the disturbing body were to revolve round it with the proper relative motion. If the moon be supposed to move in the plane of the equator, the hour- angle from the meridian of the canal may be denoted by nt + e, and if 6 be the co-latitude of any point P on the canal, we find cos ^ = sin 0.cos(w£ + e) (1) The equation of motion is therefore dt* ~ G c 2 ^ 2 -i/sin20.{l+cos2(n* + e)}. ...(2) Solving, we find c 2 ff V = -i.H r cos20-i -2 2-2 cos2 0- cos2 <y + e ) ( 3 ) 182-184] Tides in a Finite Canal 271 The first term represents a permanent change of mean level to the extent 97 = -i#cos2<9 (4) The fluctuations above and below the disturbed mean level are given by the second term in (3). This represents a semi-diurnal tide ; and we notice that if, as in the actual case of the earth, c be less than na, there will be high water in latitudes above 45°, and low water in latitudes below 45°, when the moon is in the meridian of the canal, and vice versa when the moon is 90° from that meridian. These circumstances would be all reversed if c were greater than na. When the moon is not on the equator, but has a given declination, the mean level, as indicated by the term corresponding to (4), has a coefficient depending on the declination, and the consequent variations in it indicate a fortnightly (or, in the case of the sun, a semi-annual) tide. There is also introduced a diurnal tide whose sign depends on the declination. The reader will have no difficulty in examining these points, by means of the general value of ft given in the Appendix. 184. In the case of a uniform canal encircling the globe (Arts. 181, 182) there is necessarily everywhere exact agreement (or exact opposition) of phase between the tidal elevation and the forces which generate it. This no longer holds, however, in the case of a canal or ocean of limited extent. Let us take for instance the case of an equatorial canal of finite length*. Neglecting the moon's declination we have, if the origin of time be suitably chosen, ?H^-/ 8in2 <"' + *> w with the condition that £ = at the ends, where </> = + a, say. If we neglect the inertia of the water the term dPg/dt 2 is to be omitted and we find f = \J\ \ sin 2nt cos 2a + * CO s 2nt sin 2a - sin 2 (nt + <j>)l . . . .(2) Hence v =- ^||==J#jcos 2 (nt + <j>) - 5||? cos 2nt\ , (3) where H = af/g, as in Art. 180. This is the elevation on the (corrected) 'equilibrium' theory referred to in the Appendix to this Chapter. At the centre (<£ — 0) of the canal we have V = iHcos2nt(l-^) (4) If a be small the range is here very small, but there is not a node in the absolute * H. Lamb and Miss Swain, Phil. Mag. (6), xxix. 737 (1915). A similar enect of variable depth is discussed by Goldsbrough, Proc. Lond. Math. Soc. (2) xv. 64 (1915). 272 Tidal Waves [chap, viii sense of the term. The times of high water coincide with the transits of moon and 'anti-moon*/ At the ends <f> = + a we have i tt (/-, sin4a\ rt/ , v _l — cos4a . _, , . ) 7; = J# \ 1 1 1 — J cos 2 (ratf ± a) + - sin 2 (n£ + a) [ = %HR cos 2 (nt± a + e ), ;.(5) •e r> o ^ sin 4a ^ . 1- cos 4a ,1 11 it cos2eo = l 7 — ' iioSin2e = 7 (6) Here e denotes the hour-angle of the moon W. of the meridian when there is high water at the eastern end of the canal, or E. of the meridian when there is high water at the western end. When a is small we have E =2a, 6 =-i7r + fa, (7) approximately. When the inertia of the water is taken into account we have b 4 (m 2 -l)c 2 L sin4raa l 7 vr / — sin 2 (wi — a) sin 2m (</> — a)} , (8) where m = na/c. Hence f — i-JL ^7=-f m 2 — 1 cos 2 (wi + 6) — : — - — fsin 2 (nt + a) cos 2m (<f> 4- a) ^ sm4ma l r 7 — sin 2 (n£ — a) cos 2m ((/> — a)} . .., (9) If we imagine m to tend to the limit we obtain the formula (3) of the equilibrium theory. It may be noticed that the expressions do not become infinite for m -> 1 as they would in the case of an endless canal. In all cases which are at all comparable with oceanic conditi6ns m is, however, considerably greater than unity. At the centre of the canal we have , H ( msin2a\ V = - f — 9 — =- cos 2nd 1 r— = I (10) ' 2 m 2 - 1 V sm 2ma } v ' As in the equilibrium theory, the range is very small if a be small, but there is not a true node. At the ends we find .. H ( /m sin 4a , \ _ . , N 7 ? = i-2 — T\\-r—A 1 ) cob 2 (n* ± a) z m 2 — 1 (\sin4ma / a)f = iHR 1 cos2(nt±a+ ei ), (11) if „ _ m sin 4a — sin 4ma „ . a m (cos 4ma — cos 4a) ,,„ R 1 cos 2e 2 = — — = — , , R x sin 2e x = -~-^ — , x . . . . . .(12) (m 2 - 1) sin 4ma ' (m 2 - 1) sm 4ma v ; * This term is explained in the Appendix to this Chapter, t Cf. Airy, "Tides and Waves," Art. 301. m (cos 4ma — cos 4a) . _ , . f — : , — - - sm 2 (nt ± sm 4ma 184-185] Tides in Finite Canal 273 When a is small we have R 1 =2a, 6i=-i7r + fa, (13) approximately, as in the case of the equilibrium theory. The value of R x becomes infinite when sin 4>ma = 0. This determines the critical lengths of the canal for which there is a free period equal to irjn, or half a lunar day. The limiting value of ei in such a case is given by tan 2ei = — cot 2a, or = tan 2a, according as 4ma is an odd or even multiple of it. Corrected Equilibrium Theory Dynamical Theory 2a 2aa Kange at Eange at e o Eange at Kange at «i (degrees) (miles) centre ends (degrees) centre ends (degrees) -45 -45 9 540 •004 •157 -42 •004 •165 -4V9 18 1080 •016 •311 -39 •018 •396 -38'5 27 1620 •037 •460 -36 •044 •941 -33'9 31' 5 1890 •050 •531 -34-5 •063 1-945 -30-9 36 2160 •065 •601 -33 •089 CO f -27 \ + 63 + 68-2 40 '5 2430 •081 •668 -31-6 •125 1-956 45 2700 •100 •733 -30-1 •174 •987 + 75'7 54 3240 •142 •853 -27'2 •354 •660 -83-5 63 3780 •190 •959 -24-4 •918 1-141 -65-1 72 4320 •243 1-051 -21-6 CO CO (-54 {+36 +44' 5 81 4860 •301 1-127 -18'9 1-459 1-112 90 5400 •363 1-185 -16'2 •864 •513 + 55-9 The table illustrates the case of'ra = 2*5. If 77-/^ = 12 lunar hours this implies a depth of 10820 ft., which is of the same order of magnitude as the mean depth of the ocean. The corresponding wave-velocity is about 360 sea-miles per hour. The first critical length is 2160 miles (a = ^7r). The unit in terms of which the range is expressed is the quantity H, whose value for the lunar tide is about 1-80 ft. The hour-angles e and ej are adjusted so as to lie always between ±90°, and the positive sign indicates position W. of the meridian in the case of the eastern end of the canal, and E. of the meridian for the western end. Wave-Motion in a Canal of Variable Section. 185. When the section (S, say) of the canal is not uniform but varies gradually from point to point, the equation of continuity is by Art. 169 (11), '—!&<«& •d) where b denotes the breadth at the surface. If h denote the mean depth over the width b, we have S = bh, and therefore '— js>& •(?) where h, b are now functions of x. 274 Tidal Waves [chap, viii The dynamical equation has the same form as before, viz. dt*~ 9 dx {S) Between (2) and (3) we may eliminate either rj or f ; the equation in t) is dt 2 bdxV°dx) w The laws of propagation of waves in a canal of gradually varying rect- angular section were investigated by Green*. His result, freed from the restriction to the special form of section, may be obtained as follows. If we introduce a variable t denned by 3f-&*A ■■•<•> in place of x, the equation (4) transforms into where the accents denote differentiations with respect to r. If b and h were constants, the equation would be satisfied by rj = F (r — t), as in Art. 170 ; in the present case we assume for trial, v =e.F(r-t), (7) where 9 is a function of r only. Substituting in (6), we find e' F' e" (V l h'\ (F' e'\ rt .(8) The terms of this which involve F will cancel provided a e' , b' U' A or e = Cb-$k-±, (9) C being a constant. Hence, provided the remaining terms in (8) may be neglected, the equation (4) will be satisfied. The above approximation is justified, provided we can neglect Q"/Q' and 0'/0 in com- parison with F'/F. As regards e'/6, it appears from (9) and (7) that this is equivalent to neglecting b~ l .dbjdx and h~ l .dhjdx in comparison with i}' 1 . drj/dx. If, now, X denote a wave-length, in the general sense of Art. 172, drj/dx is of the order 77/X, so that the assump- tion in question is that Xdb/dx and Xdh/dx are small compared with b and A, respectively. In other words, it is assumed that the transverse dimensions of the canal vary only by small fractions of themselves within the limits of a wave-length. It is easily seen, in like manner, that the neglect of 0"/6' in comparison with F'/F implies a similar limitation to the rates of change of dbjdx and dh/dx. Since the equation (4) is unaltered when we reverse the sign of £, the complete solution, subject to the above restrictions, is l-&-*A-t{F(r-0+/(r+0}. (10) where F and / are arbitrary functions. The first term in this represents a wave travelling in the direction of .^-positive ; the velocity of propagation past any point is determined by the consideration that any particular phase is recovered when fir and bt have equal values, and is therefore equal to *J(gh), by * "On the Motion of Waves in a Variable Canal of small depth and width," Gamb. Trans, vi. (1837) [Papers, p. 225]; see also Airy, "Tides and Waves," Art. 260. 185-186] Canal of Varying Section 275 (5), as we should expect from the case of a uniform section. In like manner the second term in (10) represents a wave travelling in the direction of ^-negative. In each case the elevation of any particular part of the wave alters, as it proceeds, according to the law The reflexion of a progressive wave at a point where the section of a canal suddenly changes has been considered in Art. 176. The formulae there given shew, as we should expect, that the smaller the change in the dimensions of the section, the smaller will be the amplitude of the reflected wave. The case where the change from one section to the other is con- tinuous, instead of abrupt, has been investigated by Rayleigh for a special law of transition*. It appears that if the space within which the transition is completed be a moderate multiple of a wave-length there is practically no reflexion; whilst in the opposite extreme the results agree with those of Art. 176. If we assume, on the basis of these results, that when the change of section within a wave-length may be neglected a progressive wave suffers no appreciable disintegration by reflexion, the law of amplitude easily follows from the principle of energy |. It appears from Art. 174 that the energy of the wave varies as the length, the breadth, and the square of the height, and it is easily seen that the length of the wave, in different pafts of the canal, varies as the corresponding velocity of propagation, and therefore as the square root of the mean depth. Hence in the above notation, rfbh^ is constant, or which is Green's law above found. 186. In the case of simple harmonic motion, where t] oc cos {at + e), the equation (4) of the preceding Art. becomes I&KlH-o a> Some particular cases of considerable interest can be solved with ease. 1°. For example, let us take the case of a canal whose breadth varies as the distance from the end # = 0, the depth being uniform ; and let us suppose that at its mouth {x=a) the canal communicates with an open sea in which a tidal oscillation t) = C cos (<rt + e) (2) is maintained. Putting h = const., b <x x, in (1), we find S-4l+* 2 -°> v provided k 2 = a 2 /gh (4) Hence "- c W) coa{,Tt+e) (5) * " On Reflection of Vibrations at the Confines of two Media between which the Transition is gradual," Proc. Lond. Math. Soc. (1) xi. 51 (1880) [Papers, i. 460]; Theory of Sound, 2nd ed., London, 1894, Art. 148 b. t Rayleigh, I.e. ante p. 260. 276 Tidal Waves [CHAP. VIII The curve y=«/ i x ) i s figured on p. 286 ; it indicates how the amplitude of the forced oscillation increases, whilst the wave-length is practically constant, as we proceed up the canal from the mouth. 2°. Let us suppose that the variation is in the depth only, and that this increases uniformly from the end x—Ooi the canal to the mouth, the remaining circumstances being as before. If, in (1), we put h — h^x\a, K = (r 2 a/gh 0i we obtain s(49+"=o- m The solution of this which is finite for #=0 is f KOI) IC 3S \ 7? = ^l- T2 + T 2- y2 -...j, (7) or i7 = ^Le/ (2#c*a?*), (8) whence finally, restoring the time-factor and determining the constant, J (2k* at) .(9) 0- The annexed diagram of the curve y = Jo{<Jx\ where, for clearness, the scale adopted for y is 200 times that of x, shews how the amplitude continually increases, and the wave-length diminishes, as we travel up the canal. These examples may serve to illustrate the exaggeration of oceanic tides which takes place in shallow seas and in estuaries. 3°. If the breadth and depth both vary as the distance from the end # = 0, we have, writing b = \x\a, h — h x/a, dx* + ^ + ^ = 0, ox .(10) where k = a 2 a/gh as before. Hence * =A i 1 ~ O + 17170" -) cos (<rt+t) (11) The series is equal to J x (2k*#*)/kz#z, and the constant A is determined by com- parison with (2). The present assumption gives a fair representation of the case of the Bristol Channel, and the tides observed at various stations are found to be in good agree- ment with the formula*. We add one or two simple problems of free oscillations. * G. I. Taylor, Gamb. Proc. xx. 320 (1921). 186] Canal of Varying Section 277 4°. Let us take the case of a canal of uniform breadth, of length 2a, whose bed, slopes uniformly from either end to the middle. If we take the origin at one end, the motion in the first half of the canal will be determined, as above, by n =AJ (2 K M), (12) where K = o- 2 a/gh , h denoting the depth at the middle. It is evident that the normal modes will fall into two classes. In the first of these 77 will have opposite values at corresponding points of the two halves of the canal, and will therefore vanish at the centre (x = a). The values of 0- are then determined by </ (2*£a*) = 0, (13) viz. < being any root of this, we have „J&£.. {Ka) k (14) a K In the second class, the value of 77 is symmetrical with respect to the centre, so that dr)/dx=Q at the middle. This gives J '(2 K M)=0 (15) It appears that the slowest oscillation is of the asymmetrical class, and corresponds to the smallest root of (13), which is 2k* a* = -765577-, whence 2tt ^ 4a — = 1*306 x 5-. 5°. Again, let us suppose that the depth of the canal varies according to the law h = h (l-^, (16) where x now denotes the distance from the middle. Substituting in (1), with 6=const., we find d_ dx {0-5)&} + »'- (17) If we put 0-2 = ^(^+1)^0^ (18) this is of the same form as the general equation of zonal harmonics, Art. 84 (1). In the present problem n is determined by the condition that 77 must be finite for x\a— +1. This requires (Art. 85) that n should be integral; the normal modes are therefore of the type v =CP n (I) . cos (at + e), (19) where P n is a zonal harmonic, the value of o- being determined by (18). In the slowest oscillation (n=l), the profile of the free surface is a straight line. For a canal of uniform depth k , and of the same length (2a), the corresponding value of <r would be 7rc/2a, where c-—(gh )i. Hence in the present case the frequency is less, in the ratio 2 ,/2tt, or -9003*. The forced oscillations due to a uniform disturbing force Xs=/cos(o-* + e) (20) * For extensions, and applications to the theory of 'seiches' in lochs, see Chrystal, "Some Results in the Mathematical Theory of Seiches," Proc. R. S. Edin. xxv. 328 (1904), and Trans. E. S. Edin. xli. 599 (1905). For more recent investigations see Proudman, Proc. Lond. Math. Soc. (2) xiv. 240 (1914); Doodson, Trans. R. S. Edin. lii. 629 (1920); Jeffreys, M. N. R. A. S., Geophys. Suppt. i. 495 (1928). 278 Tidal Waves [chap, viii can be obtained by the^rule of Art. 168 (14). The equilibrium form of the free surface is evidently i7 = ^#cos(o-* + e), (21) and, since the given force is of the normal type 7i = l, we have 1 9^-° 2 l<r<?) where o- 2 = 2gh /a 2 . 9=T7rTb^N* cos (**+«), (22) Waves of Finite Amplitude. 187. When the elevation rj is not small compared with the mean depth h, waves, even in an uniform canal of rectangular section, are no longer propagated without change of type. The question was first investigated by Airy*, by methods of successive approximation. He found that in a pro- gressive wave different parts will travel with different velocities, the wave- velocity corresponding to an elevation t) being given approximately by Art. 175(6). A more complete view of the matter can be obtained by a method similar to that adopted by Riemann in treating the analogous problem in Acoustics. (See Art. 282.) The sole assumption on which we are now proceeding is that the vertical acceleration may be neglected. It follows, as explained in Art. 168, that the horizontal velocity may be taken to be uniform over any section of the canal. The dynamical equation is du du drj /n . s* +u to = -^' (1 > as before, and the equation of continuity, in the case of a rectangular section, is easily seen to be !> + '>«> — l! < 2 > where h is the depth. This may be written _ +u ___ ( A + , )g - (o) Multiplying this equation by/' (rj), where /(??) is a function to be deter- mined, and adding to (1), we have |+«|){/«+«i— (fc+t)/w£-*g = _ (h + ,)/(,) A [/(,) + «}, (4) provided (* + *) If (I)}*- 0- * I.e. ante p. 267. 186-187] Waves of Finite Amplitude 279 This is satisfied by /(i,)-2*{(l + j[)*-l}, (5) where Cq = y/(gh). Hence, writing P=f(r,) + u, Q=f( V )-u, (6) we have f + (" + ">S= ' ••••• w and, by similar steps, *+<— >g-°. - < 8 > where v = (h + v )f\ v ) = c (l + f) ( 9 ) It appears, therefore, that P is constant for a geometrical point moving in the positive direction of x with the velocity c ,(l + ff + u, (10) whilst Q is constant for a point moving in the negative direction with the velocity <*{*+$-« (11) Hence any given value of P travels forwards, and any given value of Q travels backwards, with the velocities given by (10) and (11) respectively. The values of P and Q are determined by those of rj and u, and conversely. As an example, let us suppose that the initial disturbance is confined to the space for which a < x < b, so that P and Q are initially zero for x < a and x > b. The region within which P differs from zero therefore advances, whilst that within which Q differs from zero recedes, so that after a time these regions separate, and leave between them a space within which P = 0, Q = 0, and the fluid is therefore at rest. The original disturbance has now been resolved into two progressive waves travelling in opposite directions. In the advancing wave we have Q = 0, iP=» = 2c„|(l + g i -l} (12) so that the elevation ana the particle-velocity are connected by a definite relation (cf. Art. 171). The wave-velocity is given by (10) and (12), viz. it is Co h) 31+1-2 , (13) To the first order ofrj/h, this is in agreement with Airy's result quoted on p. 262. Similar conclusions can be drawn in regard to the receding wave*. * The above results can also be deduced from the equation (3) of Art. 173, by a method due to Earnshaw ; see Art. 283. 280 Tidal Waves [chap, viii Since the wave-velocity increases with the elevation, it appears that in a progressive wave-system the slopes will become continually steeper in front, and more gradual behind, until at length a state of things is reached in which we are no longer justified in neglecting the vertical acceleration. As to what happens after this point we have at present no guide from theory; observa- tion shews, however, that the crests tend ultimately to curl over and break. The case of a ' bore,' where there is a transition from one uniform level to another, may be investigated by the artifice of steady motion (Art. 175). If Q denote the volume per unit breadth which crosses each section in unit time we have u 1 h l = u 2 h 2 =Q, (14) where the suffixes refer to the two uniform states, h x and h 2 denoting the depths. Con- sidering the mass of fluid which is at a given instant contained between two cross-sections, one on each side of the transition, we see that in unit time it gains momentum to the amount pQ (u 2 — «i), the second section being supposed to lie to the right of the first. Since the mean pressures over the sections are \gph x and \gph 2 , we have Q(u 2 -u l ) = y(k 1 *-h 2 *) (15) Hence, and from (14), £ 2 =%W^i + A 2 ) (16) If we impress on everything a velocity — u x we get the case of a wave invading still water with a velocity of propagation -Vfr} (l7 > in the negative direction. The particle-velocity in the advancing wave is % - u 2 in the direction of propagation. This is positive or negative according as h 2 ^ h u i.e. according as the wave is one of elevation or depression. The equation of energy is however violated, unless the difference of level be regarded as infinitesimal. If, in the steady motion, we consider a particle moving along the surface stream-line, its loss of energy in passing the place of transition is \pW-u?)+gp(h,-h 2 ) (18) per unit volume. In virtue of (14) and (16) this takes the form gp{h 2 -hf Ah x h 2 (iy) Hence, so far as this investigation goes, a bore of elevation (h 2 > h x ) can be propagated unchanged on the assumption that dissipation of energy takes place to a suitable extent at the transition. If however h 2 <h x , the expression (19) is negative, and a supply of energy would be necessary. It follows that a negative bore of finite height cannot in any case travel unchanged*. 188. In the detailed application of the equations (1) and (3) to tidal phenomena, it is usual to follow the method of successive approximation. As an example, we will take the case of a canal communicating at one end {x = 0) with an open sea, where the elevation is given by rj = a cos at (20) * Kayleigh, "On the Theory of Long Waves and Bores," Proc. Roy. Soc. A, xc. 324 (1914) [Papers, vi. 250]. 187-188] Tides of Second Order 281 For a first approximation we have du dt = -9 h dx' drj_ ,du dt~ dx ,(21) .(22) the solution of which, consistent with (20), is t) = a cos a- [ t — , u = — i For a second approximation we substitute these values of rj and u in (1) and (3), and obtain du_ drj g 2 aa 2 dt~~ 9 dx~ ~~ ' 2c 3 <-?)• '»*'('-!). i-»s-^ ! -«'('-?) : •••« Integrating these by the usual methods, we find, as the solution consistent with (20), rj = a cos alt— 9<ra' c 3 x sin 2o- (<-?)• ga / x\ , a 2 a 2 _ / x\ ..g 2 aa 2 . _ ■/ ■' a?' w = ^- cos o-f ? — J- ^^-3-cos2o-( *--)- |^— T -^sin2o- f *-- 54) The annexed figure shews, with, of course, exaggerated amplitude, the profile of the waves in a particular case, as determined by the first of these equations. It is to be noted that if we fix our attention on a particular point of the canal, the rise and fall of the water do not take place symmetrically, the fall occupying a longer time than the rise. The occurrence of the factor x outside trigonometrical terms in (24) shews that there is a limit beyond which the approximation breaks down. The condition for the success of the approximation is evidently that gaax/c 3 should be small. Putting c 2 =gh, X = 27rc/o-, this fraction becomes equal to 2n- (a/h) . (x/X). Hance however small the ratio of the original elevation (a) to the depth, the fraction ceases to be small when x is a sufficient multiple of the wave-length (X). It is to be noticed that the limit here indicated is already being overstepped in the right-hand portions of the figure; and that the peculiar features which are beginning to shew themselves on the rear slope are an indication rather of the imperfections of the analysis than of any actual property of the waves. If we were to trace the curve further, we should find a secondary maximum and minimum of elevation developing themselves on the rear slope. In this way Airy attempted to explain the phenomenon of a double high- water which is observed in some rivers; but, for the reason given, the argument cannot be sustained*. The same difficulty does not necessarily present itself in the case of a canal closed by a fixed barrier at a distance from the mouth, or, again, in the case of the forced waves due to a periodic horizontal force in a canal closed at both ends (Art. 179). Enough has, however, been given to shew the general character of the results to be expected in such cases. For further details we must refer to Airy's treatise f. When analysed, as in (24), into a series of simple-harmonic functions of the time, the expression for the elevation of the water at any particular place (x) consists of two terms? * McCowan, I.e. ante p. 259. f "Tides and Waves," Arts. 198, .. Britann. (9th ed.) xxiii. 362, 363 (1888). and 308. See also G. H. Darwin, "Tides," Encyc. 282 Tidal Waves [chap, vm of which the second represents an ' over- tide,' or 'tide of the second order,' being propor- tional to a 2 ; its frequency is double that of the primary disturbance (20). If we were to continue the approximation we should obtain tides of higher orders, whose frequencies are 3, 4, ... times that of the primary. If, in place of (20), the disturbance at the mouth of the canal were given by £ = a cos crt + a' cos (a't + c), it is easily seen that in the second approximation we should in like manner obtain tides of periods 27rl(cr + a-') and 2n7((r- </) ; these are called 'compound tides.' They are analogous to the 'combination- tones' in Acoustics which were first investigated by Helmholtz*. Propagation in Two Dimensions. 189. Let us suppose, in the first instance, that we have a plane sheet of water of uniform depth h. If the vertical acceleration be neglected, the horizontal motion will as before be the same for all particles in the same vertical line. The axes of x, y being horizontal, let u, v be the component horizontal velocities at the point (x, y), and let f be the corresponding elevation of the free surface above the undisturbed level. The equation of continuity may be obtained by calculating the flux of matter into the columnar space which stands on the elementary rectangle 8x8y ; thus we have, neglecting terms of the second order, l x (uUy)^ + l y (vh&x) 8y - -| {(?+ h) 8x8y}, whence I=- A (S + |) w The dynamical equations are, in the absence of disturbing forces, du _ dp dv _ dp where we may write if zq denote the ordinate of the free surface in the undisturbed state. We thus obtain du 8f dv d£ /ox dt = -!>te> di=- g dy (2) If we eliminate u and v, we find where c 2 = gk as before. In the application to simple-harmonic motion, the equations are shortened if we assume a complex time-factor e i{<rt+e) , and reject in the end, the t "Ueber Combination stone," Berl. Monatsber. May 22, 1856 [Wiss. Abh. i. 256]; and "Theorie der Luftschwingungen in Rohren mit offenen Enden," Crelle, lvii. 14 (1859) [Wiss. Abh. i. 318]. p dt dx } p dt dy' i88-i9o] Waves on an Open Sheet of Water 283 imaginary parts of our expressions. This is legitimate so long as we have to deal solely with linear equations. We have then, from (2), u Jlf, v Jlf (4) <t dec a- oy whilst (3) becomes S + p + ^=° (5) where A^o^/c 2 (6) The condition to be satisfied at a vertical bounding wall is obtained at once from (4), viz. it is !=°> < 7 > if 8n denote an element of the normal to the boundary. When the fluid is subject to small disturbing forces whose variation within the limits of the depth may be neglected, the equations (2) are replaced by du = _ 9£_9ft ^ = _ d l_ d & /ft\ dt~ 9 dx dx* dt~ 9 dy dy' {) where Q is the potential of these forces. If we put ?=-0/#, (9) so that f denotes the equilibrium-elevation corresponding to the potential XX these may be written ir-4<^>- l-4 (? - ?) (10) In the case of simple-harmonic motion; these take the forms -£s«-& -?£«-& • < X1 > whence, substituting in the equation of continuity (1) we obtain (V 1 2 + ^)ir=v 1 ^ ) (i2) if v *-b*k (13) and P = a 2 /gh, as before. The condition to be satisfied at a vertical boundary is now 4ff-f>'- < 14) 190. The equation (3) of Art. 189 is identical in form with that which presents itself in the theory of the transverse vibrations of a uniformly stretched membrane. A still closer analogy, when regard is had to the boundary-conditions, is furnished by the theory of cylindrical waves of sound*. Indeed many of the results obtained in this latter theory can be at once transferred to our present subject. * Eayleigh, Theory of Sound, Art. 338. g84 Tidal Waves [chap, viii Thus, to find the free oscillations of a sheet of water bounded by vertical walls, we require a solution of ov+m-o, (i) subject to the boundary-condition I- » Just as in Art. 178 it will be found that such a solution is possible only for certain values of k, which accordingly determine the periods (2-Tr/kc) of the various normal modes. Thus, in the case of a rectangular boundary, if we take the origin at one corner, and the axes of x, y along two of the sides, the boundary- conditions are that 3f/3# = for x — and x = a, and 3f/3y = for 2/ = and y = b, where a, 6 are the lengths of the edges parallel to x, y respectively. The general value of f subject to these conditions is given by the double Fourier's series f=22-4 m>n cos — — cos-yS (3) where the summations include all integral values of m, n from to oo . Substituting in (1) we find »--(3!+-J) < 4 > If d > 6, the component oscillation of longest period is got by making m = 1, ft = 0, whence ka = tt. The motion is then everywhere parallel to the longer side of the rectangle. Cf. Art. 178. 191. In the case of a circular sheet of water, it is convenient to take the origin at the centre, and to transform to polar co-ordinates, writing x = r cos 0, y = r sin 6. The equation (1) of the preceding Art. becomes This might of course have been established independently. As regards dependence on 0, the value of f may, by Fourier's Theorem, be supposed expanded in a series of cosines and sines of multiples of 0; we thus obtain a series of terms of the form /(r) C0S l J v ; sinj •(2) It is found on substitution in (1) that each of these terms must satisfy the equation independently, and that f(T) + lf (r) + (*-J)/(r)-0 (3) i90-i9i] Circular Basin 285 This is of the same form as Art. 101 (14). Since f must be finite for r = 0, the various normal modes are given by Z=A s J 8 (kr) C °^ s6 .cos(<rt + e), (4) where s may have any of the values 0, 1, 2, 3, ..., and A 8 is an arbitrary constant. The admissible values of h are determined by the condition that d£/dr = at the boundary r = a, say, or J s '(ka) = (5) The corresponding ' speeds ' (cr) of the oscillations are then given by <r = kc, where c = \/(gh). In the case 5 = 0, the motion is symmetrical about the origin, so that the waves have annular ridges and furrows. The lowest roots of J '(&a) = 0, or Ji(A;a) = 0, (6) are given by ka/7r = 1-2197, 2-2330, 3*2383, ..., (7) these numbers tending ultimately to the form &a/7r = ra + £, where m is integral *. Hence o-a/c=2-832, 7'016, 10*173, (7a) In the mth mode of the symmetrical class there are m nodal circles whose radii are given by f = or J (kr) = (8) The roots of this are*f" &r/7r=-7655, 1*7571, 2'7546, (9) For example, in the first symmetrical mode there is one nodal circle r = *628a. The form of the section of the free surface by a plane through the axis of z, in any of these modes, will be understood from the drawing of the curve y = Jo (#)> which is given on the next page. When s > there are s equidistant nodal diameters, in addition to the nodal circles J s (kr) = (10) It is to be noticed that, owing to the equality of the frequencies of the two modes represented by (4), the normal modes are now to a certain extent indeterminate ; viz. in place of cos sd or sin sd we might substitute cos s (0 — a s ), where a s is arbitrary. The nodal diameters are then given by a 2m +1 0-&s= — 27" 71 "' ( n ) where ra = 0, 1, 2, ..., 5 — 1. The indeterminateness disappears, and the frequencies become unequal, if the boundary deviate, however slightly, from the circular form. * Stokes, "On the Numericai Calculation of a class of Definite Integrals and Infinite Series." Camb. Trans, ix. (1850) [Papers, ii. 355]. It is to be noticed that ka/v is equal to tJt, where r is the actual period, and r is the time a progressive wave would take to travel with the velocity *J(gh) over a space equal to the diameter 2a. f Stokes, I.e. 286 Tidal Waves [chap, viii 191J Circular Basin 287 In the case of the circular boundary, we obtain by superposition of two fundamental modes of the same period, in different phases, a solution Z=C s J s (kr).cos(crt + s6 + €) (12) This represents a system of waves travelling unchanged round the origin with an angular velocity ajs in the positive or negative direction of 6. The motion of the individual particles is easily seen from Art. 189 (4) to be elliptic-harmonic, one principal axis of each elliptic orbit being along the radius vector. All this is in accordance with the general theory recapitulated in Art. 168. The most interesting modes of the unsymmetrical class are those corre- sponding to s = 1, e.g. ^AJiikr) cos 0. cos (crt + e), (13) where k is determined by J 1 '(ka) = (14) The roots of this are * kalw= '586, 1-697, 2-717,..., (15) whence aa/c =1*841, 5'332, 8536, (15a) We have now one nodal diameter (6 = %7r), whose position is, however, in- determinate, since the origin of 6 is arbitrary. In the corresponding modes for an elliptic boundary, the nodal diameter would be fixed, viz. it would coincide with either the major or the minor axis, and the frequencies would be unequal. The diagrams on the next page shew the contour-lines of the free surface in the first two modes of the present species. These lines meet the boundary at right angles, in conformity with the general boundary-condition (Art. 190 (2)). The simple-harmonic vibrations of the individual particles take place in straight lines perpendicular to the contour-lines, by Art. 189 (4). The form of the sections of the free surface by planes through the axis of z is given by the curve y = Ji {x) on the opposite page. The first of the two modes here figured has the longest period of all the normal types. In it, the water sways from side to side, much as in the slowest mode of a canal closed at both ends (Art. 178). In the second mode there is a nodal circle, whose radius is given by the lowest root of J x (kr) = ; this makes r— '7l9af. * See Eayleigh's treatise, Art. 339. A general formula for calculating the roots of J 8 ' (ka) = Q, due to Prof. J. M c Mahon, is given in the special treatises. t The oscillations of a liquid in a circular basin of any uniform depth were discussed by Poisson, " Sur les petites oscillations de l'eau contenue dans un cylindre," Ann. de Gergonne, xix. 225 (1828-9) ; the theory of Bessel's Functions had not at that date been worked out, and the results were consequently not interpreted. The full solution of the problem, with numerical details, was given independently by Eayleigh, Phil. Mag. (5), i. 257 (1876) [Papers, i. 25]. The investigation in the text is limited, of course, to the case of a depth small in comparison with the radius a. Poisson's and Eayleigh's solution for the case of finite depth will be noticed in Chapter ix. 288 Tidal Waves [chap. VIII i9i] Properties of BesseVs Functions 289 A comparison of the preceding investigation with the general theory of small oscilla- tions referred to in Art. 168 leads to several important properties of Bessel's Functions. In the first place, since the total mass of water is unaltered, we must have /2ir fa o J {rdOdr = 0, (16) where £ has any one of the forms given by (4). For s > this is satisfied in virtue of the trigonometrical factor cos s6 or sin s6 ; in the symmetrical case it gives / J (kr)rdr=0 (17) o Again, since the most general free motion of the system can be obtained by super- position of the normal modes, each with an arbitrary amplitude and epoch, it follows that any value whatever of £, which is subject to the condition (16), can be expanded in a series of the form (= 22 (A, cos s6 + B 8 sin s6)J 8 (kr), (18) where the summations embrace all integral values of s (including 0) and, for each value of 5, all the roots k of (5). If the coefficients A 8 , B 8 be regarded as functions of t, the equa- tion (18) may be regarded as giving the value of the surface-elevation at any instant. The quantities A 8 , B 8 are then the normal co-ordinates of the present system (Art. 168) ; and in terms of them the formulae for the kinetic and potential energies must reduce to sums of squares. Taking, for example, the potential energy v=isrp!iC*d*<ty, (19) /2tt ra o jo this requires that / / w 1 w 2 rd6dr=0, (20) where w u w 2 are any two terms of the expansion (18). If w it w 2 involve cosines or sines of different multiples of 0, this is verified at once by integration with respect to 6 ; but if we take w x oc J 8 (k x r) cos s0, w 2 « e/g far) cos s6, where k x , k 2 are any two distinct roots of (5), we get J 8 (k x r)J 8 (k 2 r)rdr = (21) /; o The general results, of which (17) and (21) are particular cases, are /: 'J (kr)rdr= -^J '(ka) (22) (cf. Art. 102 (10)), and / J* far) J 8 far) rdr= _ {k 2 aJ 8 ' faa) J 8 fad) - k x aJ 8 ' fad) J 8 fad)}. ...(23) In the case of k x = k 2 the latter expression becomes indeterminate; the evaluation in the usual manner gives / a {Js(^)Yrdr = ~[Pa^{J 8 '(ka)^-h(k^-s^{J 8 (ka)Y] (24) For the analytical proofs of these formulae we refer to the treatises cited on p. 136. The small oscillations of an annular sheet of water bounded by concentric circles are easily treated, theoretically, with the help of Bessel's Functions of the second kind/ The only case of any special interest, however, is when the two radii are nearly equal ; we then have practically a re-entrant canal, and the solution follows more simply by the method of Art. 178. 290 Tidal Waves [chap, viii The analysis can also be applied to the case of a circular sector of any angle*, or to a sheet of water bounded by two concentric circular arcs and two radii. An approximation to the frequency of the slowest mode in an elliptic basin of uniform depth can be obtained by Rayleigh's method, referred to in Art. 168. The equation of the boundary being a. + fc" 1 -* (25) let us assume, for the component displacements, (26) *-H SM>. n a 2 ' where the constants have been adjusted so as to make 3*?-* <w at the boundary (25). The time-factor cos at is understood. The corresponding surface- elevation is f~»(g+g)-s<"+*>* .(28) The assumption (26) is however too general for the present purpose, since it includes circulatory motions. The condition of zero vorticity requires (2a 2 + b 2 )B=2a 2 A (29) We find from (26) 2T=phj[(£ 2 + r} 2 ) dxdy = 2npabho- 2 Ua 2 + t \AB + h-^ + ^^j B*l sin 2 at, ...(30) 2V=gp (U 2 dxdy=2nabgh 2 .2^^ cos 2 at (31) Expressing that the mean value of T — V is zero, and introducing the relation (29), we find „ 18a 2 + 6& 2 c 2 5a 2 + 26 2 'a 2 ' ■(32) where c 2 =gh. If we put b = a, this makes aa/c = 1*852, the true value for the circular basin being 1*841. The approximate estimate is in excess, in accordance with a general principle (Art. 168). The various modes of longitudinal oscillations in an elliptic canal have been studied by Jeffreys "j* and Goldstein J, and more recently by Hidaka§, by different methods. It appears that in the gravest mode o-ajc = 1 '8866, whilst if we make bja-^0 in (32) we get a-a/c = 1 '8994. It would appear that the formula gives a good approximation for values of bja less than unity. * See Rayleigh, Theory of Sound, Art. 339. f Proc. Lond. Math. Soc. (2) xxiii. 455 (1924). + Ibid, xxviii. 91 (1927). § Mem. Imp. Mar. Obs. (Japan), iv. 99 (1931). This paper includes the discussion of the free oscillations in basins with boundaries of various other shapes, and with various laws of depth. 191-193] Basin of Variable Depth 291 192. As an example of forced oscillations in a circular basin, let us suppose that the disturbing forces are such that the equilibrium elevation would be ?=<) cos s6 . cos (at + e) (33) This makes ^^=0, so that the equation (12) of Art. 189 reduces to the form (1), above, and the solution is %= A J s (kr) cos sd. cos (at + e), (34) where A is an arbitrary constant. The boundary-condition (Art. 189 (14)) gives AkaJg (ka) = sG, q T ( ]cf\ whence ? = G , //,» \ cos s6 . cos (at + e) (35) The case s = 1 is interesting as corresponding to a uniform horizontal force ; and the result may be compared with that of Art. 179. From the case s — 2 we could obtain a rough representation of the semi- diurnal tide in a polar basin bounded by a small circle of latitude, except that the rotation of the earth is not as yet taken into account. We notice that the expression for the amplitude of oscillation becomes infinite when J/ (ka) = 0. This is in accordance with a general principle, of which we have already had several examples ; the period of the disturbing force being now equal to that of one of the free modes investigated in the preceding Art. 193*. When the sheet of water is of variable depth, the calculation at the beginning of Art. 189 gives, as the equation of continuity, 9r_ d(hu) d(hv) dt dx dy K } The dynamical equations (Art. 189 (2)) are of course unaltered. Hence, eliminating f, we find, for the free oscillations, dt 2 If the time-factor be e i(<Tt+ *\ we obtain -»&(»£K(*©I < 2 > £(»S+4«K<" < 3 > dx When h is a function of r, the distance from the origin, only, this may be written ^+fl#=° w As a simple example we may take the case of a circular basin which shelves gradually from the centre to the edge, according to the law A = Ao ( 1 ~5) (5) * This formed Art. 189 of the 2nd ed. of this work (1895). A similar investigation was given by Poincar^, Legons de mecanique celeste, iii. 94 (Paris, 1910). 292 Tidal Waves [chap, viii Introducing polar co-ordinates, and assuming that £ varies as cos s6 or sin s6. the equation (4) takes the form \ l a 2 )\dr 2 + rdr r 2 V a* r dr + ghj °* (6) That integral of this equation which is finite at the origin is easily found in the form of an ascending series. Thus, assuming t=2A, y > & where the trigonometrical factors are omitted, for shortness, the relation between consecu- tive coefficients is found to be (m 2 -s 2 )A m = \m(m-2)-s 2 -^-i A m _ 2 , a 2 a 2 or, if we write — j— =n (n— 2)-s 2 , (8) where n is not as yet assumed to be integral, (m 2 -s 2 ) A m =(m-n) (m + n-Z) A m _ 2 ( 9 ) The equation is therefore satisfied by a series of the form (7), beginning with the term A 8 (rja) 8 , the succeeding coefficients being determined by putting m = s + 2, s+4, ... in (9). We thus find t- A f-Yll - ( n - s -2)( n + s ) ^ , (n-s-4) (n-s-2) (n + s) (n + s + 2) r*__ \ Q ~ 8 \a)\ 2(2s + 2) a 2+ 2. 4 (2s + 2) (2s + 4) a 4 "'J ' { } or in the usual notation of hypergeometric series {=A.?.F(a,f},y,£) (11) where a=i»i+Js, ^ = l+^s-^n, y=s + l. Since these make -y-a — /3 = 0, the series is not convergent for r=a, unless it terminate. This can only happen when n is integral, of the form s ■+- 2j. The corresponding values of o- are then given by (8). In the symmetrical modes (s = 0) we have C= A <>Y I2~^ + W72 2 ^"-J' {l2) where j may be any integer greater than unity*. It may be shewn that this expression vanishes for^' — 1 values of r between and a, indicating the existence of j— 1 nodal circles. The value of o- is given by o*-VU"l)*pP. 03) whence o-a/ N /(^ ) = 2 * 828 5 4 ' 899 > 6 ' 928 > ( 13a ) The gravest symmetrical mode (j=-2) has a nodal circle of radius '707 a. Of the unsymmetrical modes, the slowest, for any given value of 5, is that for which 7i=s + 2, in which case we have £= A 8 — cos s0 cos (a-t + e). a» the value of o- being given by <r 2 =2s ,~ (14) 9 k o In the case s=l the various frequencies are given by ^=(4/ 2 -2)4°, (is; a' whence aa/Jfaho) = 1-414, 3'742, 5-831, (16) * If we put r/a.= sin £x> the series is identical with the expansion of -Py-i (cos x) 5 see Art. 85 (4). 193-194] BesseTs Function of the Second Kind 293 In the slowest of these modes, corresponding to s=l, w=3, the free surface is always plane. It appears from Art. 191 (15 a) that the frequency is '768 of that of the corre- sponding mode in a circular basin of uniform depth A , and of the same radius* As in Art. 192 we could at once write down the formula for the tidal motion produced by a uniform horizontal periodic force ; or, more generally, for the case where the disturbing potential is of the type Q, oc r s COS s6 COS (at + e). 194. We may conclude this discussion of 'long' waves on plane sheets of water by an examination of the mode of propagation of disturbances from a centre in an unlimited sheet of uniform depth. For simplicity, we will consider only the case of symmetry, where the elevation £ is a function of the distance r from the origin of disturbance. This will introduce us to some peculiar and rather important features which attend wave-propagation in two dimensions. The investigation of a periodic disturbance involves the use of a Bessel's Function (of zero order) 'of the second kind,' as to which some preliminary notes may be useful. To solve the equation -=-£ + - -^- + <£ = (1) by definite integrals, we assume f cf>= l e~ zt Tdt, •(2) where T is a function of the complex variable t, and the limits of integration are constants as yet unspecified. This makes dty d<j> Z dz* by a partial integration. The equation (1) is accordingly satisfied by t-jwik' (3) provided the expression J{ 1 + 1 2 ) e ~ zt vanishes at each limit of integration. Hence, on the supposition that z is real and positive, or at all events has its real part positive, the integral in (3) may be taken along a path joining any two of the points t, —i, +oo in the plane of the variable t ; but two distinct paths joining the same points will not necessarily give the same result if they include between them one of the branch -points (t= ±i) of the function under the integral sign. Thus, for example, we have the solution e-^dt *-/-, N/(i+* 2 r where the path is the portion of the imaginary axis which lies between the limits, and that value of the radical is taken which becomes =1 for t=0. If we write t = g + ir), we obtain /l e~ izr >dr} n*" —j— ^ =2i I cos (zcosS) d$ = iirJ (z), (4) -W(l-»7 2 ) Jo which is the solution already met with (Art. 1 00). * For the oscillations in an elliptic basin with a similar law of depth see Goldsbrough, Proc. Roy. Soc. A, cxxx. 157 (1930). f Forsyth, Differential Equations, c. vii. The systematic application of this method to the theory of Bessel's Functions is due to Hankel, "Die Cylinderfunktionen erster u. zweiter Art," Math. Ann. i. 467 (1869). 294 Tidal Waves [chap, viii An independent solution is obtained if we take the integral (3) along the axis of rj from the point (0, i) to the origin, and thence along the axis of £ to the point (oo , 0). This gives, with the same determination of the radical, d{irj) Z* 00 e~^d^ _ f 00 e~*d% . /"I e-***d v fO e -i*Vd(iri) .("> e -*d£; _ f ,(5) By adopting other pairs of limits, and other paths, we can obtain other forms of <j), but these must all be equivalent to fa or fa, or to linear combinations of these. In particular, some other forms of fa are important. It is known that the value of the integral (3) taken round any closed contour which excludes the branch- points (t = ± i) is zero. Let us first take as our contour a rectangle, two of whose sides coincide with the positive portions of the axes of | and rj, except for a small semi- circular indentation about the point t = i, whilst the remaining sides are at infinity. It is easily seen that the parts of the integral due to the infinitely distant sides will vanish, either through the vanishing of the i ^)— factor e~ z % when £ is infinite, or through the infinitely rapid fluctuation of the function e~ iz1 >jr} when n is in- finite. Hence for the path which gave us (5) we may substitute that which extends along the axis of n from the point (0, i) to (0, i oo ), provided the continuity of the radical be attended to. Now as the variable t travels counter-clockwise round the small semicircle, the radical changes continuously from J ( 1 - ?7 2 ) to i J (rj 2 — 1 ). We have therefore .(6) *■-/< H/(^ 2 -l)"ii sW-irJo 6 dU It will appear that this solution is the one which is specially appropriate to the case of diverging waves. Another method of obtaining it will be given in Chapter x. If we equate the imaginary parts of (5) and (6) we obtain 2 f 00 Jq{ z )~ -\ sm ( z cosa u ) du, (7) 7TJ o a form due to Mehler*. On account of the physical importance of the solution (6) it is convenient to have a special notation for it. We write f D (z) = - \ e - iecoshu du (8) "■y o This is equivalent to D (z)= — Y (z) — iJ (z), (9) where J F (z) = — — \ cos (z cosh u) du (10) IT J Equating the real parts of (5) and (6) we have, also, 2 f 00 2 f$ n Y«(z)=--\ e-* sinhu du + - sin (z cos S) dS (11) tr J o "J * Math. Ann. v. (1872). f The use of a simple notation to meet the case of diverging waves seems justifiable. Our D (z) is equivalent to - iH W (z) in Nielsen's notation, as slightly modified by Watson. J This is the notation definitely recommended by Watson. The reader should be warned, however, that the same symbol has been employed by other writers in various senses. From a purely mathematical point of view the choice of a standard solution 'of the second kind' is largely a matter of convention, since the differential equation (1) is still satisfied if we add any constant multiple of J (z). Tables of the function Y (z) as defined by (10) are given in Watson's treatise. 194] Asymptotic Expansion 295 For a like reason, the path adopted for <j> 2 may be replaced by the line drawn from the point (0, i) parallel to the axis of £ (viz. the dotted line in the figure). To secure the con- tinuity of jj(l + t 2 ), we note that as t describes the lower quadrant of the small semicircle, the value of the radical changes from J (I -rf-) to e* ln V(2£), approximately. Hence along the dotted line we have, putting *=i+£, V(l+0=e*"V(2£-^ 2 ), where that value of the radical is to be chosen which is real and positive when £ is infini- tesimal. Thus (12) p +< ^ rf tf + Q 1 _^ g) f^ ^ V2 Ji **V(2£-^ 2 ) ^ 2 io $ K igj g If wc expa^u Jie binomial, and integrate term by term, we find ™-®'<™W@+ 1J £(ff+-~}> w where use has been made of the formulae z* z* (14) .-ztrm-kj? H (*~B _ 1 ■ 3 ... (2m- 1) tt* If we separate the real and imaginary parts of (13) we have, on comparison with (9), Mz)=(^- Z ) {Bsm(z + i7r)-Scos(z + %7r)}, (15) Fo( * )= ~Gl) { EG0S ( z +fr)+ S *™( z +i«)}, .(16) 1 2 .3 2 1 2 .3 2 .5 2 .7 2 ^ where R =i -__ + __^ _,.,, ^ l 2 1 2 .3 2 .5 2 I (1 ' ll(8i) 3!(82) 3+- -- J The series in (13) and (17) are of the kind known as 'semi-convergent,' or 'asymptotic,' expansions ; i.e. although for sufficiently large values of z the successive terms may for a while diminish, they ultimately increase again indefinitely, but if we stop at a small term we get an approximately correct result*. This may be established by an examination of the remainder after m terms in the process of evaluation of (12). It follows from (15) that the large roots of the equation J (z) = approximate to those of sin(z + J;7r) = (18) The series in (13) gives ample information as to the demeanour of the function D (z) when z is large. When z is small, D Q (z) is very great, as appears from (8). An approxi- mate formula for this case can be obtained as follows. Keferring to (11), we have /;^-*.^--(-i)*-/;^{x +5£+ i^ + ...}* =/;^>^(S-> ™ * Cf. Whitfcaker and Watson, Modern Analysis, c. viii.; Bromwich, Theory of Infinite Series, London, 1908, c. xi. ; Watson, c. vii. ; Gray and Mathews, c. iv. The semi-convergent expansion of J (z) is due to Poisson, Journ. de VEcole Polyt. cah. 19, p. 349 (1823); a rigorous investigation of this and other analogous expansions was given by Stokes, I.e. ante p. 285. The 'remainder' was examined by Lipschitz, Crelle, lvi. 189 (1859). Cf. Hankel, I.e. ante p. 293. 296 Tidal Waves [chap, viit The first term gives* /QO Q-W dw=-y-\og\z+., .(20) and the remaining ones are small in comparison. Hence, by (9) and (11), D (s)=--(log^ + y+^7r + ...) (21) IT It follows that lim«Z> '(«)=--t (22) The formula (21) is sufficient for our purposes, but the complete expression can now be obtained by comparison with the general solution of (1) in terms of ascending series, viz. J f Z 2 2 4 2 6 1 <t> = AJ (z) + B y (z)\ogz + - 2 -s 2 ^-^ 2 + s 3 227427^2- •••[ .(23) where n m = l +^ + 3 + — + -• In order to identify this with (21), for small values of z, we must make B=--, 4=--(log^+ 7 +i*V) (24) 7T 7T Hence 2 2 (2 2 S 4 2 6 ) D (z)=--(\og fe+y+frV) J (z)~- |p-%22— p+ g 3 g2 ; 42 ; 62 " -J • - (25) 195. We can now proceed to the wave-problem stated at the beginning of Art. 194. For definiteness we will imagine the disturbance to be caused by a variable pressure p applied to the surface. On this supposition the dynamical equations near the beginning of Art. 1 89 are replaced by du _ _ 3f _ 1 dp dv _ _ 1 9f _ 1 dp m dt dx p dx dt pdy p dy ' I=-4M) ( 2 > as before. If we introduce the velocity-potential in(l), we have, on integration, 8-* + ? v We may suppose that p refers to the change of pressure, and that the arbi- trary function of t which has been incorporated in <£ is chosen so that d(j>/dt = in the regions not affected by the disturbance. Eliminating f by means of (2), we have S=^+;t <*> When (f> has been determined, the value of f is given by (3). * De Morgan, Differential and Integral Calculus, London, 1842, p. 653. t The Bessel's Functions of the second kind were first thoroughly investigated and made available for the solution of physical problems in an arithmetically intelligible form by Stokes, in a series of papers published in the Camb. Trans. With the help of the modern Theory of Functions, some of the processes have been simplified by Lipschitz and others, and (especially from the physical point of view) by Kayleigh. These later methods have been used in the text. I Forsyth, Differential Equations, c. vi. note 1; Watson, Bessel Functions, pp. 59, 60. 194-195] Waves Diverging from a Centre 297 We will now assume that p is sensible only over a small* area about the origin. If we multiply both sides of (4) by SocSy, and integrate over the area in question, the term on the left-hand side may be neglected (relatively), and we find d i ds= jphjt\[t* dxd y' (5) where 8s is an element of the boundary of the area, and 8n refers to the hori- zontal normal to 8s, drawn outwards. Hence the origin may be regarded as a two-dimensional source, of strength /«-*** (6) where P is the total disturbing force. Turning to polar co-ordinates, we have to satisfy g_,g*. + !g*), (7) or \3r 2 r dr/ where c 2 = gh, subject to the condition lim (_fcrg)-/ W (8) where f(t) is the strength of the source, as above defined. In the case of a simple-harmonic source e^ the equation (7) takes the form ^1+^ = 0. W where k = a/c, and a solution is (t> = iD (kr)e^, (10) where the constant factor has been determined by Art. 194(22). Taking the real part we have (p=i {Jo (kr) sin at — Y (kr) cos at}, (11) corresponding to f (t) = cos at. For large values of kr the result (10) takes the form i iff [ t-- j -\iir *=v(8^r (12) The combination t — r/c indicates that we have, in fact, obtained the solution appropriate to the representation of diverging waves. It appears that the amplitude of the annular waves ultimately varies inversely as the square root of the distance from the origin. * That is, the dimensions of the area are small compared with the 'length' of the waves generated, this term being understood in the general sense of Art. 172. On the other hand, the dimensions must be supposed large in comparison with h. 298 Tidal Waves [chap, viii 196. The solution we have obtained for the case of a simple-harmonic source e* * may be written /•oo itrit- - cosh u ] 2tt</)=| e V c y ck (13) This suggests generalization by Fourier's Theorem ; thus the formula 27r<£=| fit — coshujdu (14) should represent the disturbance due to a source f(t) at the origin*. It is implied that the form of fit) must be such that the integral is convergent ; this condition will as a matter of course be fulfilled whenever the source has been in action only for a finite time. A more complete formula, embracing both converging and diverging waves, is 2tt(/)= fU — ^coshujdu+j F (t 4- - cosh u J du (15) The solution (15) may be verified, subject to certain conditions, by substitution in the differential equation (7). Taking the first term alone, we find = / jsinh 2 u.f" (t — cosh u) — cosh u.f (t — cosh u H du =—,\ n/U — cosh u ) du= — sinh u . f [ t — cosh u ) r'Jo du* J \ c J r{_ J \ c J_\ u =o This obviously vanishes whenever f(t)=0 for negative values of t exceeding a certain limit f. Again — 27r?--^ = -/ coshw./'U — cosh u\du — - j (sinh u + e~ u )f U--cosh?tj du — -I / U~- cosh u\\ +-/ e~ u f (t--coshu \du =/ ( < -3 + "c/o 00e "" / ('"o- cosh K ) A under the same condition. The limiting value of this when r^-0 is f (t) ; and the state- ment made above as to the strength of the source in (14) is accordingly verified. A similar process will apply to the second term of (15) provided F (t) vanishes for positive values of t exceeding a certain limit. 197. We may apply (14) to trace the effect of a temporary source varying according to some simple prescribed law. If we suppose that everything is quiescent until the instant t — 0, so that * The substance of Arts. 196, 197 is adapted from a paper "On Wave -Propagation in Two Dimensions," Proc. Lond. Math. Soc. (1), xxxv. 141 (1902). A result equivalent to (14) was obtained (in a different manner) by Levi-Civita, Nuovo Cimento (4), vi. (1897). t The verification is very similar to that given by Levi-Civita. 196-197] Waves Diverging from a Centre 299 f{t) vanishes for negative values of t, we see from (14) or from the equivalent form 2 ^ri ag (i6) that </> will be zero everywhere so long as t < r/c. If, moreover, the source acts only for a finite time r, so that f(t) = for t > t, we have, for t > r + r/c, rr /(fl)rffl (17) . This expression does not as a rule vanish ; the wave accordingly is not sharply defined in the rear, as it is in front, but has, on the contrary, a sort of 'tail'f whose form, when t — r/c is large compared with t, is determined by 2 ^=^T7^/ T /(*><** < 18 > The elevation f at any point is given by (3), viz. *-W (19) It follows that /: £eft=0, (20) provided the initial and final values of cf> vanish. It may be shewn that this will be the case when/(£) is finite and the integral f • J -00 /(*)* (21) is convergent. The meaning of these conditions appears from (6). It follows that even when dP /dt is always positive, so that the flux of liquid in the neighbourhood of the origin is altogether outwards, the wave which passes any point does not consist solely of an elevation (as it would in the corre- sponding one-dimensional problem) but, in the simplest case, of an elevation followed by a depression. To trace in detail the progress of a solitary wave in a particular case we may assume /»-*£?■ (22 > which makes P increase from one constant value to another according to the law ^=.4+5 tan- 1 - (23) * Analytically, it may be noticed that the equation (4), when_p = 0, may be written gy ay ay _ n daP^dfp^diict?' and that (17) consists of an aggregate of solutions of the known type {xt + yz + iictf}-*. t The existence of the ' tail ' in the case of cylindrical electric waves was noted by Heaviside, Phil. Mag. (5), xxvi. (1888) [Electrical Papers, ii.]. 300 Tidal Waves [chap, viii The disturbing pressure has now no definite epoch of beginning or ending, but the range of time within which it is sensible can be made as small as we please by diminishing r. For purposes of calculation it is convenient to assume /«-,- ..(24) in place of (22), and to retain in the end only the imaginary part. We have then dz 2tt(£ = f 00 du fi dz I t-- coshu-iT I t ir-(t-\ it) z 2 J o c J o c \ c J .(25) where z = tanh hi. We now write Z-ir^cfo-v* t -r--iT=b 2 e- 2 ^, .(26) where we may suppose that a, b are positive, and that the angles a, /3 lie between and ^n. Since --H + r 2, b * = { t + L)+ T 2 tan 2a = , tan 28 ct — r .(27) ct + r' it appears that a < b according as t^ 0, and that a > /3 always. With this notation, we find 2ti-0 = 2 I 1 - 2 e ~2ia_f ) 2 e -2ip z 2 ab .(28) z " e -i(a-d) To interpret the logarithms, let us mark, in the plane of a complex variable z, the points o Since the integral in the second member of (28) is to be taken along the path 01, the proper value of the third member is <*(« + £) ab {(i* g+f. op/)- (kg |$-<. <*/)}, where real logarithms and positive values of the angles are to be understood. Hence, rejecting all but the imaginary part, we find «*-*i£ffli*S + =§fcs<.-«» .(29) as the solution corresponding to a source of the type (22). Here IP _ ( a 2 + 2ab cos (a- fl) + 6 2 \£ 2a6sin(a-/3) IQ V-2a6cos(a-/3) + W ' tan/ ^" 52^2 and the values of a, b, a, |8 in terms of r and £ are to be found from (27). ,(30) 197-198] Solitary Wave 301 It will be sufficient to trace the effect of the most important part of the wave as it passes a point whose distance r from the origin is large compared with cr. If we confine ourselves to times at which t - rjc is small compared with rjc, a. will be small compared with b, PIQ will be a small angle, and IPjIQ will = 1, nearly. If we put T t=- + T tan?;, (31) we shall have *=i*-h> a=V(rsec79), j8=icr/r, 6-(2r/c)*, (32) approximately ; and the formula (29) will reduce to 2 ^ = ^ C0Sa= ^W cos d 7r -h)>/( cos n) (33) ! The elevation £ is then given by 2 "^=^j(^r^(vf s ™^-M cosi '> < 34 > approximately. The diagram shews the relation between £ and t, as given by this formula*. 198. We proceed to consider the case of a spherical sheet, or ocean, of water covering a solid globe. We will suppose for the present that the globe does not rotate, and we will also in the first instance neglect the mutual attraction of the particles of the water. The mathematical conditions of the question are then exactly the same as in the acoustical problem of the vibrations of spherical layers of air*f*. Let a be the radius of the globe, h the depth of the fluid; we assume that h is small compared with a, but not (as yet) that it is uniform. The position of any point on the sheet being specified by the angular co-ordinates 0, <f), let u be the component velocity of the fluid at this point along the meridian, in the direction of 6 increasing, and v the component along the parallel of latitude, in the direction of <£ increasing. Also let f denote the elevation of the free surface above the undisturbed level. The horizontal * The points marked - 1, 0, + 1 correspond to the times rjc - r, rjc, r/c + r, respectively, f Discussed in Eayleigh's Theory of Sound, c. xviii. 302 Tidal Waves [chap, viii motion being assumed, for the reasons explained in Art. 172, to be the same at all points in a vertical line, the condition of continuity is 4 (uha sin 686) 86 + ^- (vha 86) 8cf> = - a sin 68<f> . a86 . |f , otf ocp 01 where the left-hand side measures the flux out of the columnar space standing on the element of area a sin 68(f) . a 86, whilst the right-hand member expresses the rate of diminution of the volume of the contained fluid, owing to fall of the surface. Hence 3? 1_ \ d(hu sin 6) d(hv)\ dt~ asin6\ d6 d<j> J W If we neglect terms of the second order in u, v, the dynamical equations are, on the same principles as in Arts. 169, 189, dt~ 9 add ad6' dt~ g asm6d<l> asm6d<t>' where fl denotes the potential of the extraneous forces. If we put Z = -n/g, (3) these may be written dt~ ad6 K * Ui dt asm6d<f>^ *' w Between (1) and (4) we can eliminate u, v, and so obtain an equation in f only. Iu the case of simple-harmonic motion, the time-factor being e i{<Tt+e) , the equations take the forms r _ % (d (hu sin 6/) d(hv)\ ,„. *~<rasm0( ~W " + ~df]' W -<&»«-& — "sifo^^ft (6) 199. We will now consider more particularly the case of uniform depth. To find the free oscillations we put ? = ; the equations (5) and (6) of the preceding Art. then lead to ^— - Q — n sintf^j + ^-o-5 5tI+ —r ? = (1) sin 6d6\ d6J sin 2 6d(f> 2 gh & v ' This is identical in form with the general equation of spherical surface- harmonics (Art. 83 (2)). Hence, if we put <7 2 a 2 ^r=^ +1 >' < 2 > a solution of (1) will be ? = S n , (3) where S n is the general surface-harmonic of order n. It was pointed out in Art. 86 that # n will not be finite over the whole sphere unless n be integral. Hence, for an ocean covering the whole globe, 198-199] Waves on a Spherical Ocean 303 the form of the free surface at any instant is, in any fundamental mode, that of a 'harmonic spheroid* r = a + h + S n cos (at + e), (4) and the speed of the oscillation is given by , -{»(» + l))i. <2*? (5) Co the value of n being integral. The characters of the various normal modes are best gathered from a study of the nodal lines (S n = 0) of the free surface. Thus, it is shewn in treatises on Spherical Harmonics * that the zonal harmonic P n (/jl) vanishes for n real and distinct values of /jl lying between + 1, so that in this case we have n nodal circles of latitude. When n is odd one of these coincides with the equator. In the case of the tesseral harmonic the second factor vanishes for n — s values of fi, and the trigonometrical factor for 2s equidistant values of <f>. The nodal lines therefore consist of n— s parallels of latitude and 2s meridians. Similarly the sectorial harmonic (l-^)& C0S \n<t> smj has as nodal lines 2n meridians. These are, however, merely special cases, for since there are 2n + 1 inde- pendent surface-harmonics of any integral order n, and since the frequency, determined by (5), is the same for each of these, there is a corresponding degree of indeterminateness in the normal modes, and in the configuration of the nodal lines. We can also, by superposition, build up various types of progressive waves ; e.g. taking a sectorial harmonic we get a solution in which f oc (1 - /a 2 )*** cos (w0 - <r* + €) ; (6) this gives a series of meridianal ridges and furrows travelling round the globe, the velocity of propagation, as measured at the equator, being aa /n+ 1\£ ?-Pf •<** (7) It is easily verified, on examination, that the orbits of the particles are now ellipses having their principal axes in the directions of the meridians and parallels, respectively. At the equator these ellipses reduce to straight lines. In the case n = 1, the harmonic is always of the zonal type. The harmonic spheroid (4) is then, to our order of approximation, a sphere excentric to the globe. It is important to remark, however, that this case * For references see p. 110. 304 Tidal Waves [chap, viii is, strictly speaking, not included in our dynamical investigation, unless we imagine a constraint applied to the globe to keep it at rest; for the de- formation in question of the free surface would involve a displacement of the centre of mass of the ocean, and a consequent reaction on the globe. A corrected theory for the case where the globe is free could easily be investigated, but the matter is hardly important, first because in such a case as that of the earth the inertia of the solid globe is so enormous compared with that of the ocean, and secondly because disturbing forces which can give rise to a deformation of the type in question do not as a rule present themselves in nature. It appears, for example, that the first term in the expression for the tide-generating potential of the sun or moon is a spherical harmonic of the second order (see the Appendix to this Chapter). When n = 2, the free surface at any instant is approximately ellipsoidal. The corresponding period, as found from (5), is then '816 of that belonging to the analogous mode in an equatorial canal (Art. 181). For large values of n the distance from one nodal line to another is small compared with the radius of the globe, and the oscillations then take place much as on a plane sheet of water. For example, the velocity of propagation, at the equator, of the sectorial waves represented by (6) tends with increasing n to the value (gh)%, in agreement with Art. 170. From a comparison of the foregoing investigation with the general theory of Art. 168 we are led to infer, on physical grounds alone, the possibility of the expansion of any- arbitrary value of £ in a series of surface -harmonics, thus o the coefficients of the various independent harmonics being the normal co-ordinates of the system. Again, since the products of these coefficients must disappear from the expressions for the kinetic and potential energies, we are led to the ' conjugate ' properties of spherical harmonics quoted in Art. 87. The actual calculation of the energies will be given in the next Chapter, in connection with an independent treatment of the same problem. The effect of a simple-harmonic disturbing force can be written down at once from the formula (14) of Art. 168. If the surface value of O be expanded in the form n = xa n (8) where H w is a surface-harmonic of integral order n, the various terms are normal components of force, in the generalized sense of Art. 135 ; and the equilibrium value of f corresponding to any one term X2 B is Cn-.-Q.lg (9) Hence, for the forced oscillation due to this term, we have &— i-is^r 1 < 10) l—o- j<T n g i99-2oo] Free and Forced Oscillations 305 where a measures the 'speed' of the disturbing force, and a n that of the corresponding free oscillation, as given by (5). There is no difficulty, of course, in deducing (10) directly from the equations of the preceding Art. 200. We have up to this point neglected the mutual attraction of the parts of the liquid. In the case of an ocean covering the globe, and with such relations of density as we meet with in the actual earth and ocean, this is not insensible. To investigate its effect in the case of the free oscillations, we have only to substitute for X2 n , in the last formula, the gravitation - potential of the displaced water. If the density of this be denoted by p, whilst p represents the mean density of the globe and liquid combined, we have* _ jiTTvpa n ~ 2n + l U ' (11) and g =%y7rap , (12) 7 denoting the gravitation-constant, whence fl »=-2irh>v^ < 13 > Substituting in (10) we find 2H-2^)> ^ where cr n is now used to denote the actual speed of the oscillation, and <r n ' the speed calculated on the former hypothesis of no mutual attraction. Hence the corrected speed is given by 3 p\gh 2n + 1 p 0/ For an ellipsoidal oscillation {n = 2), and for p/p = '18 (as in the case of the Earth), we find from (14) that the effect of the mutual attraction is to lower the frequency in the ratio of *94 to 1. The slowest oscillation would correspond to n = 1, but, as already indicated, it would be necessary, in this mode, to imagine a constraint applied to the globe to keep it at rest. This being assumed, it appears from (15) that if p> po the value of a x 2 is negative. The circular function of t is then replaced by real exponentials; this shews that the configuration in which the surface of the sea is a sphere concentric with the globe is one of unstable equilibrium. Since the effect of the constraint is merely to increase the inertia of the system, we infer that the equilibrium is still unstable when the globe is free. In the extreme case where the globe itself is supposed to have no gravitative * See, for example, Eouth, Analytical Statics, 2nd ed., Cambridge, 1902, ii. 146-7. t This result was given by Laplace, Mecanique Celeste, Livre l er , Art. 1 (1799). The free and the forced oscillations of the type n = 2 had been previously investigated in his "Eecherches sur quelques points du systeme du monde," Mem. de V Acad. roy. des Sciences, 1775 [1778] [Oeuvres Completes, ix. 109, ...]. '- i -(-+i)(i-5^i3S < l5 >t 306 Tidal Waves [chap, viii power at all, it is obvious that the water, if disturbed, would tend ultimately, under the influence of dissipative forces, to collect itself into a spherical mass, the nucleus being expelled. It is obvious from Art. 168, or it may easily be verified independently, that the forced vibrations due to a given periodic disturbing force, when the gravitation of the water is taken into account, will be given by the formula (10), provided Xl n now denote the potential of the extraneous forces only, and cr n have the value given by (15). 201. The oscillations of a sea bounded by meridians, or parallels of latitude, or both, can also be treated by the same method*. The spherical harmonics involved are however, as a rule, no longer of integral order, and it is accordingly difficult to deduce numerical results. In the case of a zonal sea bounded by two parallels of latitude, we assume t={Ap(ri + Bq(ri} C ™\scl>, (1) whence .(3) sin, where /t=cos 6, and p (/x), q (it) are the two functions of it, containing (1 - it 2 )£ 8 as a factor, which are given by the formula (2) of Art. 86. It will be noticed that p (it) is an even, and q (fi) an odd function of p. If we distinguish the limiting parallels by suffixes, the boundary conditions are that u=0 for fi=fii and n = fi2' For the free oscillations this gives, by Art. 198 (6), Ap'M + Bq'fa^O, Ap'(n 2 ) + Bq'(fji 2 ) = 0, (2) p' M, 9.' M p' M, q' M which is the equation to determine the admissible values of n, the order of the harmonics- The speeds (<r) corresponding to the various roots are given as before by Art. 199 (5). If the two boundaries are equidistant from the equator, we have /x 2 = — /*i", The above solutions then break up into two groups ; viz. for one of these we have b=o, yw=o, (4) and for the other .4=0, q' (i^) = (5) In the former case £ has the same value at two points symmetrically situated on opposite sides of the equator ; in the latter the values at these points are numerically equal, but opposite in sign. If we imagine one of the boundaries to be contracted to a point (say /x 2 = 1), we pass to the case of a circular basin. The values of p' (1) and q' (1) are infinite, but their ratio can be evaluated by means of formulae given in Art. 84. This gives, by the second of equations (2), the ratio A : B, and substituting in the first we get the equation to determine n. A simpler method of treating this case consists, however, in starting with a solution which is known to be finite, whatever the value of n, at the pole it = l. This involves a change of variable, as to which there is some latitude of choice. We might take, for instance, the expression for P n 8 (cos 6) in Art. 86 (6), and seek to determine n from the condition that ^P n »(cos0) = O (6) for = 6^. By making the radius of the sphere infinite, we can pass to the plane problem of Art. 191 J. The steps of the transition will be understood from Art. 100. * Cf. Rayleigh, I.e. ante p. 301. + This question has been discussed by Macdonald, Proc. Lond. Math. Soc. xxxi. 264 (1899^ J Cf. Rayleigh, Theory of Sound, Arts. 336, 338. 200-203] Waves on a Limited Ocean 307 If the sheet of water considered have as boundaries two meridians (with or without parallels of latitude), say = and = o, the condition that v=0 at these restricts us to the factor cossco, and gives sa = mir, where m is integral. This determines the admissible values of s, which are not in general integral*. The diurnal and semi-diurnal tides in a non-rotating ocean of uniform depth bounded by two meridians have been studied by Proudman and Doodson, and worked out for special cases and for special depths f. Dynamics of a Rotating System. 202. The theory of the tides on an open sheet of water is seriously complicated by the fact of the earth's rotation. If, indeed, we could assume that the periods of the free oscillations, and of the disturbing forces, were small compared with a day, the preceding investigations would apply as a first approximation, but these conditions are far from being fulfilled in the actual circumstances of the earth. The difficulties which arise when we attempt to take the rotation into account have their origin in this, that a particle having a motion in latitude tends to keep its angular momentum about the earth's axis unchanged, and so to alter its motion in longitude. This point is of course familiar in connection with Hadley's theory of the trade-winds {. Its bearing on tidal theory seems to have been first recognized by Maclaurin§. Owing to the enormous inertia of the solid body of the earth compared with that of the ocean, the effect of tidal reactions in producing periodic changes of the angular velocity is quite insensible. This angular velocity will therefore for the present be treated as constant. The theory of the small oscillations of a dynamical system about a state of equilibrium relative to a real or ideal rigid frame which rotates with con- stant angular velocity about a fixed axis differs in some important particulars from the theory of small oscillations about a state of absolute equilibrium, of which some account was given in ilrt. 168. It is therefore worth while to devote a little space to it before entering on the consideration of special problems. The system considered may be entirely free, or it may be connected with a rotating solid. In the latter case it is assumed that the connecting forces as well as the internal forces of the system are subject to the 'con- servative' law. 203. The equations of motion of a particle m relative to rectangular axes Ox, Oy, Oz which rotate about Oz with angular velocity co are m (x - 2coy — a> 2 x) — X, m(y+2'*)x—<o 2 y)=Y, mz=Z, ...(1) where X, Y, Z are the impressed forces. * The reader who wishes to carry the study of the problem further in this direction is referred to Thomson and Tait, Natural Philosophy (2nd ed.), Appendix B, " Spherical Harmonic Analysis." f M. N. R. A. S., Geophy. Suppt. i. 468 (1927), and ii. 209 (1929). % "The Cause of the General Trade Winds," Phil. Trans. 1735. § De Causd Physicd Fluxus et Refluxus Maris, Prop, vii.: "Motus aquee turbatur ex insequali velocitate qua corpora circa axem Terrse motu diurno deferuntur" (1740). 308 Tidal Waves [chap, viii Let us now suppose that the relative co-ordinates {%, y, z) of each particle are expressed in terms of a certain number of independent quantities q 1} q 2 , ... q r . We write ^jSm^ + ^ + i 2 ), T =|o) 2 2m(^ + 2/ 2 ) (2) Hence ® denotes the kinetic energy of the relative motion, which we shall suppose expressed as a homogeneous quadratic function of the generalized velocities q r , with coefficients which are functions of the generalized co- ordinates q r ; whilst T is the kinetic energy of the system when rotating, without relative motion, in the configuration (q lt q 2 , ... q n ). Finally we put 2 (X&*+ YBy 4- ZBz) = -8V+ Q 1 8q 1 + Q 2 8q 2 + ... + Q n 8q ni ...(3) where Fis the potential energy and Q lt Q 2i ... Q n are the generalized com- ponents of extraneous force. If we multiply the three equations (1) by dx/dq r , dy/dq r) dz/dq r , respec- tively, and add, and sum the result for all the particles of the system, and then proceed as in the 'direct' proof of Lagrange's equations, we obtain the following typical equation of motion in generalized co-ordinates * : 5f-£+At* + A-* + - + A-*'— 4 (F - W + *' - (4) n K Of 1/) where ft r8 = 2o>Sm - , ' % (5) It is to be noted that /3r S = -/3sr, /3 rr =0 (6) The equation (4) may also be derived from Art. 141 (23), with the help of Art. 142 (8), by supposing the rotating frame to be free, but to have an infinite moment of inertia. The conditions for relative equilibrium, in the absence of disturbing forces, are found by putting q 1} q 2y ... q r = in (4), whence a!< F - r °) = ' < 7 > shewing that the equilibrium value of V— T is 'stationary.' Again, from (1) we have Sw (xx + yy + zz) — a> 2 2m {xx + yy + zz) = 1 (Xx + Yy + Zz), . . .(8) or, by (2) and (3) j t ('&+V-To) = Q 1 q 1 + Q*qz+. ~ + Qnq n (9) This result may also be deduced from (4), taking account of the relations (6). * Cf. Thomson and Tait, Natural Philosophy (2nd ed.), i. 310; Lamb, Higher Mechanics , 2nded., Art. 84. 203-204] Dynamics of a Rotating System 309 When there are no disturbing forces we have ®+ F-r = const (10) The form assumed by the Hamiltonian theorem of Art. 135 is also to be noticed. The total kinetic energy of our system is T=$2m\(a;- coy) 2 + {y + coxf + i 2 } = ® + T + coM, (11) where M=Xin(xy — yx) (12) If there are no extraneous forces we have JV V)dt = 0, (13) subject to the usual terminal condition. Hence a[ (® + T a + ©if- V)dt=0, (14) with the condition Xm {(x — coy) Ax + (y + ax) Ay 4- zAz}\ =0 (15) Jto This theorem may also be deduced directly from (1) by the usual Hamiltonian procedure, and leads in turn t6 an independent proof of the equations (4), for the case of free motion. The inclusion of disturbing forces in the investigation presents no difficulty. The condition (15) is fulfilled whenever the initial and final relative con- figurations are the same in the varied as in the actual motion. 204. We will now suppose the co-ordinates q r to be chosen so as to vanish in the undisturbed state. In the case of a small disturbance, we may then write 2®< = a n q 1 2 + a 22 q<?+ ... +2a 12 q t q 2 + ... , (1) 2(F-T )=c 11 gi 2 + c 22 g 2 2 -f-...+2c 12 9^ 2 + ..., (2) where the coefficients may be treated as constants. The terms of the first degree in V — T have been omitted, on account of the 'stationary' property. In order to simplify the equations as much as possible, we will further suppose that, by a linear transformation, each of these expressions is reduced, as in Art. 168, to a sum of squares; viz. 2® = a 1 q 1 2 + o 2 q 2 2 + ... +a n g n 2 , (3) 2(V-T )=c l q 1 * + c 2 q 2 *+...+c n q 7 ? (4) The quantities q 1} q 2 , ... q n may be called the 'principal co-ordinates' of the system, but we must be on our guard against assuming that the same simplicity of properties attaches to them as in the case of no rotation. The coefficients a x , a 2 , ... a n and c 1} c 2 , ... c n may be called the 'principal co- efficients' of inertia and of stability, respectively. The latter coefficients 310 Tidal Waves [chap, vm are the same as if we were to ignore the rotation, and to introduce fictitious 'centrifugal' forces (ma> 2 x, mcoPy, 0) acting on each particle in the direction outwards from the axis. The equations (4) of the preceding Art. become, in the case of infinitely small motions, «1 qi +C X q X + #12 J2 + £l3<Z3 + • • • + $lnin = Ql, d 2 q 2 + C 2 <72 + #21<?1 + /3 2 3<?3 + . . . + fiznqn = $2, -G.J ,(5) Ctn'qn + C n q n + Pnl4l + &*&& + £ TC 3<?3 + where the coefficients j3 rs may be regarded as constants. If we multiply these by q x , q 2 , ... J n in order and add, we find, taking account of the relation /3 rs — — fi sr , d dt (® + F- T ) = QKJ! + $2^2+ ... + Qntfn, .(6) as has already been proved without approximation. 205. To investigate the free motions of the system, we put Q lf Q 2 , ... Q n = 0, in (5), and assume, in accordance with the usual method of treating linear equations, q 1 = A 1 e", q 2 = A 2 e kt , ... q n =A n e kt (7) Substituting, we find (Oi\*+Qi)ili +i8iiX4i+... + /3 ln \A n = 0, fi 21 \A 1 + (a ? \ 2 + c 2 ) A 2 + ... +/3 2n \A n = 0, /3 nX \A x + /5 n2 X^ 2 + • • • + KX 2 + o n ) A n = 0. Eliminating the ratios A X :A 2 : ... :A n , we get the equation tfiX 2 + Ci, p X2 \, ... f3 Xn \ fi 2i \, Cl 2 \ 2 + C 2 , ... /3 2 nX .(8) &l\ /3 n2 X, ... a n X 2 + c„ = 0, .(9) or, as we shall occasionally write it, for shortness, D(\)=0 (10) The determinant D (X) comes under the class called by Cayley ' skew- determinants,' in virtue of the relations (6) of Art. 203. If we reverse the sign of X, the rows and columns are simply interchanged, and the value of the determinant is therefore unaltered. Hence the equation (10) will involve only even powers of X, and the roots will be in pairs of the form X= + (p +i<r). In order that the configuration of relative equilibrium should be stable it is essential that the values of p should all be zero, for otherwise terms of the forms e ±pt cos at and e ±pe sin at would present themselves in the realized 204-205] Condition for Stability 311 expression for any co-ordinate q r . This would indicate the possibility of an oscillation of continually increasing amplitude. In the theory of absolute equilibrium, sketched in Art. 168, the necessary and sufficient condition of stability (in the above sense) was simply that the potential energy must be a minimum in the configuration of equilibrium. In the present case the conditions are more complicated*, but it is easily seen that if the expression for V— T be essentially positive, in other words if the coefficients Ci, c%, ... c n in (4) be all positive, the equilibrium must be stable. This follows at once from the equation V + (F- To) = const., (11) proved in Art. 203, which shews that under the present supposition neither ® nor V— T can increase beyond a certain limit depending on the initial circumstances f. It will be observed that this argument does not involve the use of approximate equations. Hence stability is assured if V — T is a minimum in the configuration of relative equilibrium. But this condition is not essential, and there may even be stability (from the present point of view) with V — T a maximum, as will be shewn presently in the particular case of two degrees of freedom. It is to be remarked, however, that if the system be subject to dissipative forces, however slight, affecting the relative co-ordinates q ly q%, ... q n , the equi- librium will be permanently or 'secularly' stable only if V— T is a minimum. It is the characteristic of such forces that the work done by them on the system is always negative. Hence by (6) the expression Q£ + (V — T ) will, so long as there is any relative motion of the system, continually diminish, in the algebraical sense. Hence if the system be started from relative rest in a configuration such that V — T is negative, the above expression, and therefore a fortiori the part V - T , ,vill assume continually increasing negative values, which can only take place by the system deviating more and more from its equilibrium-configuration. This important distinction between 'ordinary' or kinetic, and secular' or practical stability was first pointed out by Thomson and TaitJ. It is to be observed that the above investigation presupposes a constant angular velocity (g>) maintained, if necessary, by a proper application of force to the rotating solid. When the solid is free, the condition of secular stability takes a somewhat different form, to be referred to later (Chapter XII.). In the * They have been investigated by Kouth, I.e. ante p. 195 ; see also his Advanced Rigid Dynamics, c. vi. t The argument was originally applied to the theory of oscillations about a configuration of absolute equilibrium (Art. 168) by Dirichlet, " Ueber die Stabilitat des Gleichgewichts, "• Crelle, xxxii. (1846) [Werke, Berlin, 1889-97, ii. 3]. An algebraic proof is indicated in Higher Mechanics, 2nd ed., Art. 99. J Natural Philosophy (2nd ed.), Part i. p. 391. See also Poincar£, " Sur l'equilibre d'une masse fluide animee d'un mouvement de rotation," Acta Mathematica, vii. (1885), and op. cit. ante p. 146. Some simple mechanical illustrations are given in a paper "On Kinetic Stability," Proc. Roy. Soc. A, lxxx. 168 (1909), and in the author's Higher Mechanics, 2nd ed., p. 253. 312 Tidal Waves [chap, vm practical applications we shall be concerned only with cases where V— T is a minimum, and the coefficients Ci, c 2 , ... c n in Art. 204 (4) accordingly positive. To examine the character of a free oscillation, in the case of stability, we remark that if X be any root of (10), the equations (8) give ^=— 2 =...=^=a, (12) «i a 2 a n where a lt a^, ... a n are the minors of any row in the determinant D (X), and G is arbitrary. These minors will as a rule involve odd as well as even powers of X, and so assume unequal values for the two oppositely signed roots (+ X) of any pair. If we put X = ± iar, the corresponding values of a r will be of the forms fi r ± iv r , where fj, r , v r are real. Hence q r = G (fj, r + iv r ) e i<rt + G' (fi r - iv r ) e~ ift K If we put G = \K&\ G' = \Ke- ie , we get a solution of our equations in real form, involving two arbitrary constants K, e; thus q r = K {fi r cos (at + e) — v r sin (at + e)} (13) This formula expresses what may be called a ' natural mode ' of oscillation of the system. The number of such possible modes is of course equal to the number of pairs of roots of (9), i.e. to the number of degrees of freedom of the system. It is to be noticed, as an effect of the rotation, that the various co-ordinates are no longer in the same phase. If £, t], £ denote the component displacements of any particle from its equilibrium position, we have . dx dx dy , dy . dz dz dz .(14) Substituting from (13), we obtain a result of the form g = P. K cos (<rt + e) + P' . K sin (<rt + e), \ V = Q. K cos (<rt + e) + Q' .K sin (<rt + e), I (15) £=R . iTcos (<rt+€) + R' . Ksin (<rt+c), J where P, P', Q, Q\ R, R' are determinate functions of the mean position of the particle, involving also the value of <r, and therefore different for the different normal modes, but independent of the arbitrary constants K, e. These formulae represent an elliptic- harmonic motion of period 27r/<r, the directions i-l-i and L-JL-A. da) being those of two conjugate semi-diameters of the elliptic orbit, of lengths (P 2 +Q 2 + R 2 )^. K, and (P' 2 + Q' 2 + R' 2 )l . K, respectively. The positions and forms and relative dimensions of the elliptic orbits, as well as the relative phases of the particles in them, are accordingly in each natural mode determinate, the absolute dimensions and epochs being alone arbitrary. 205-205 b] Free Oscillations 313 205 a. When the angular velocity co is small the normal modes will as a rule differ only slightly from the case of no rotation, and expressions for the altered types and frequencies can then be found as follows*. Since the determinantal equation (9) of Art. 205 is unaltered when we reverse the signs of all the &'s, the frequencies will usually involve these quantities in the second order. Hence, considering for example the mode in which A x is finite, whilst A 2 ,A 3) ... A n are relatively small, and writing \ = i<r ly the rth equation of the system (8) gives, approximately, A 1 _ ffirlOl m x Al a r (<Ti*-<r*y KU) where a r 2 = c r ja r . Hence, substituting in the first equation, we get a corrected value of o-] 2 ; thus <V = Ml + S - H i (18) «1 l r aidr Ol* ~ 0> 2 )J But these approximations fail if any denominator in the bracket vanishes or is even small. This case arises when two or more of the normal modes in the absence of rotation have the same or nearly the same period. Suppose, for instance, that a x 2 and a 2 2 are nearly equal. We have then, from (8), with A = icr, (ci 2 - a 2 a^ Ai + i/3 12 crA 2 = 0, •(19) ^21^1 + (c 2 - o 2 a 2 ) A 2 = 0, so that Ax and A 2 are comparable. Eliminating Ai/Ai, we have (^-^)(^-^) = §^^ (20) In the case of exact equality this gives * 2 -" 2=± v(Sr- (21) *- i - ± «^53' (22) approximately. The change of frequency due to the rotation is now proportional to co instead of w 2 . The values of A z , A i} ... A n in terms of A 1} A 2 are to be found from the remaining equations of the system (8), but would only affect the above con- clusion by terms involving co 2 . 205 b. On account of the analytical difficulties which attend the deter- mination of the free modes of oscillation, especially in the case of continuous systems, it is natural to look for an approximate method of calculating the more important frequencies, analogous to that employed by Rayleigh in the case of non-rotating systems (Art. 168). * Rayleigh, Phil. Mag. (6) v. 293 (1903) {Papers, v. 89]. 314 Tidal Waves [chap, viii For this purpose we may have recourse to the variational formula (14) of Art, 203. In the application to small oscillations it is convenient to express this in terms of the displacements (f, rj, f) of the particles from their positions of relative equilibrium. Writing oo + f, y + 77, z + f for x, y, z, where x Q , yo,z refer to the equilibrium position, we have A \ tl Mdt = A \ h M'dt+ r2m(ar ^-yoAf)T\ (1) Jt J t L Jk where M! = 2m(&- v $) (2) When the integrated terms in (1) are incorporated in the terminal con- dition (15) of Art. 203, the theorem becomes &[ tl (® + coM' + To-V)dt = O i (3) J t with the condition 2m {($ - co V ) Af + (v + »|) At; + ?Af } = (4) Let us now suppose that the varied, as well as the natural motion, is simple-harmonic with the same period 2ir/a y and that the limits of integration £0, £1 differ by an exact period. The terms in (4) which relate to the two limits will then cancel, so that the postulated condition is fulfilled. The result is that the mean value (with respect to time) of the expression W+uM'-iV-To) (5) is stationary for small arbitrary variations of the type of vibration, the period being kept constant. In terms of generalized co-ordinates (assumed to vanish in relative equi- librium) M' will be a bilinear function of the two sets of variables qi,q2> ... q n and q lt q 2i ... q ny whilst ® and V—T are already by hypothesis homogeneous quadratic functions of the velocities and co-ordinates, respectively. Hence (5) is a homogeneous quadratic function of the variables q r , q r . If we now write q r = A r cos at + B r sin at, (6) and denote the resulting mean value of the expression (5) by J, we have J=a*P + aQ-R y (7) where P, Q, R are certain homogeneous quadratic functions of the variables A r , B ry whose precise forms are not required for the moment. The stationary property asserts that <7 2 AP + <tAQ-AE = (8) for all infinitesimal values of AA r ,Ai? r . In particular, putting A^l r = e^ r , &B r =eB r , where e is an infinitesimal constant independent of r, we have ^=0, (9) 205 b] Free Periods; Approximations 315 on account of the homogeneous character. The statement that in a free oscillation the mean value of the expression (5) is zero is a generalization of a result already pointed out in the case of a> = 0, viz. that in oscillations about absolute equilibrium the mean values of the kinetic and potential energies are equal. The present result can be expressed in another form. If for a moment we regard a as a function of A r , B r , where these coefficients have general values, determined by the equation a*P + aQ-R=0, (10) we have (2o-P+Q)A<r + (<r 2 AP + o-AQ-A£)==0 (11) Hence when A r , B r have the special values appropriate to a free mode of oscillation, we have A<r = 0, (12) by (8). In other words, the values of a determined by (10) are stationary. It follows that if the values of P, Q, R in (10) are calculated on the basis of an assumed type of vibration which differs slightly from the truth, the error in the consequent values of a will be of the second order. These stationary values will include, as generally most important, the maxima and minima (in absolute value) of a. Applications of the above principle to particular cases will be found in Arts. 212 a, 216. The general form of the functions P, Q, R in (7) may be noticed, although it is not essential to the argument. We have at once, on reference to Art. 204 (3) (4), P = iS r a r (A r 2 + B r 2 ) } R = lS r c r {A* + B r 2 ), (13) where S r denotes a summation of terms of the types indicated, with r = 1 , 2, . . . n. Again, from (2), = \ {qxSrPlrqr + 22#r£2r?r + • • • + ?„# r /3 nr a r }, (14) where ft,-2.S«l^ (15) o{q ai q r ) Substituting from (6), and taking uhe mean value, we have Q = \8 T a,fi n A,B r , (16) where, in the double summation, each permutation of suffixes is to be taken once. As a verification we may note that if with these values of P, Q, R we form the equation (8) the coefficients of A^l r , AP r will be found to be identical with the coefficients of cos at and sin at, respectively, when we substitute from (6) in the typical equation of motion, Art. 204 (5). (01Y 316 Tidal Waves [chap, viii 206. The symbolical expressions for the forced oscillations due to a periodic disturbing force are easily written down. If we assume that Qi, Qz> • •• Qn all vary as e^ 1 , where a is prescribed, the equations (5) of Art. 204 give, if we omit the time-factors, D(i<r)q r = a rl Q 1 + a r2 Q 2 + ...+a™Qn> (1) where the coefficients on the right-hand side are the minors of the rth row in the determinant D (ia). The most important point of contrast with the theory of the 'normal modes' in the case of no rotation is that the displacement of any one type is no longer affected solely by the disturbing force of that type. As a con- sequence, the motions of the individual particles are, as is easily seen from Art. 205 (14), now in general elliptic-harmonic. Again, there are in general differences of phase, variable with the frequency, between the displacements and the force. As in Art. 168, the displacement becomes very great when D (ia) is very small, i.e. whenever the 'speed* <r of the disturbing force approximates to that of one of the natural modes of free oscillation. When the period of the disturbing forces is infinitely long, the displace- ments tend to the 'equilibrium- values' qi = Qi/ci, g 2 = Q 2 /c 2 , ... q n = Qnlc n , (2) as is seen directly from the equations (5) of Art. 204. This conclusion must be modified, however, when one or more of the coefficients of stability c 1} c 2 , ... c n is zero. If, for example, Ci = 0, the first row and column of the deter- minant D (X) are both divisible by \, so that the determinantal equation has a pair of zero roots. In other words we have a possible free motion of infinitely long period. The coefficients of Q 2 , Q3, ••• Qn on the right-hand side of (1) then become indeterminate for a — 0, and the evaluated results do not as a rule coincide with (2). This point is of importance, because in some hydrodynamical applications, as we shall see, steady circulatory motions of the fluid, with a constant deformation of the free surface, are possible when no extraneous forces act; and as a consequence forced tidal oscillations of long period do not necessarily approximate to the values given by the equi- librium theory of the tides. Cf. Arts. 214, 217. In order to elucidate the foregoing statements we may consider more in detail the case of two degrees of freedom. The equations of motion are then of the forms «i?i + cig'i+/3£ 2 = # 1 , a 2 J2 + c 2 g'2-0yi = #2 ( 3 ) The equation determining the periods of the free oscillations is ai«2X 4 + (aiC 2 + a 2 c 1 +/3 2 )X 2 +CiC 2 =0 (4) For ' ordinary ' stability it is sufficient that the roots of this quadratic in X 2 should be real and negative. Since a x , a 2 are essentially positive, it is easily seen that this condition is in any case fulfilled if c x , c 2 are both positive, and that it will also be satisfied even when 206-207] Forced Oscillations 317 0|, c 2 are both negative, provided £ 2 be sufficiently great. It will be shewn later, however, that in the latter case the equilibrium is rendered unstable by the introduction of dissipa- tive forces. See Art. 322. To find the forced oscillations when Q u Q 2 vary as e* "', we have, omitting the time- factor, (c 1 -a 2 a 1 )q 1 + iorPq 2 =Qi, - i<rpqi+(c 2 - <r 2 a 2 ) q 2 =Q 2 , (5) , onpfl n _ (02-0*02) Qi-i<rpQ 2 _ ^^i + (ci-q- 2 ai)^ 2 , fi s wnence q x - ^ _ ^ ^ _ ^ _ ^ , q 2 ^ _ ^ ^ _ ^ _^ 2 W Let us now suppose that c 2 = 0, or, in other words, that the displacement q 2 does not affect the value of V— T . We will also suppose that Q 2 =0, i.e. that the extraneous forces do no work during a displacement of the type q 2 . The above formulae then give q ^a 2 (c 1 -a 2 a 1 )+^ ^ = a 2 (c 1 -a*a 1 )+^ Ql (7) In the case of a disturbance of long period we have o-=0, approximately, and therefore a-s^s*' ^^+w Qi (8) The displacement q x is therefore less than its equilibrium -value, in the ratio 1 : 1 +/3 2 /«2<?i ; and it is accompanied by a motion of the type q 2 although there is no extraneous force of the latter type (cf. Art. 217). We pass, of course, to the case of absolute equilibrium, considered in Art. 168, by putting /3 = 0*. It should be added that the determination of the 'principal co-ordinates' of Art. 204 depends on the original forms of f& and V — T , and is therefore affected by the value of co 2 , which enters as a factor of T . The system of equations there given is accordingly not altogether suitable for a discussion of the question how the character and the frequencies of the respective principal modes of free vibration vary with co. One remarkable point which is thus overlooked is that types of circulatory motion, which are of infinitely long period in the case of no rotation, may be converted by the slightest degree of rotation into oscillatory modes of periods comparable with that of the rotation. Cf. Arts. 212, 223. To illustrate the matter in its simplest form, we may take the case of two degrees of freedom. If c 2 vanishes for o>=0, and so contains o> 2 as a factor in the general case, the two roots of equation (4) are X 2 = - c x \a x , X 2 = - c 2 /a 2 , approximately, when o> 2 is small. The latter root makes X <x co, ultimately. 207. Proceeding to the hydrodynamical examples, we begin with the case of a plane horizontal sheet of water having in the undisturbed state a motion of uniform rotation about a vertical axisf. The results will apply without serious qualification to the case of a polar or other basin, of not too great dimensions, on a rotating globe. * The preceding theory appeared in the 2nd ed. (1895) of this work. The effect of friction is considered in Art. 322. t Sir W. Thomson, "On Gravitational Oscillations of Rotating Water," Proc. R. S. Edin. x. 92 (1879) [Papers, iv. 141]. 318 Tidal Waves [chap, viii Let the axis of rotation be taken as axis of z. The axes of x and y being now supposed to rotate in their own plane with the prescribed angular velocity co, let us denote by u, v, w the velocities at time t, relative to these axes, of the particle which then occupies the position (x, y, z). The actual velocities of the same particle, parallel to the instantaneous positions of the axes, will be u — ayy, v + cox, w, and the accelerations in the same directions will be Du _ o Bv a „ Dw ^ -&*-«*, JJJ + 2.U-.V, -jj t - In the present application, the relative motion is assumed to be infinitely small, so that we may replace D/Dt by d/dt. Now let z be the ordinate of the free surface when there is relative equilibrium under gravity alone, so that 2 zo = i — (# 2 + y z ) 4- const., (1) as in Art. 26. For simplicity we will suppose that the slope of this surface is everywhere very small ; in other words, if r be the greatest distance of any part of the sheet from the axis of rotation, co 2 r/g is assumed to be small. If z +£ denote the ordinate of the free surface when disturbed, then on the usual assumption that the vertical acceleration of the water is small compared with g, the pressure at any point (x, y, z) will be given by p-Po = gp(*o+S-*)> (2) , ldp 2 3f I dp 2 dt whence ^=-co*x-g^-, --£■ = - co 2 y - g ^ . pdx a ox pdy dy The equations of horizontal motion are therefore du a? an dv i . a? an where ft denotes the potential of the disturbing forces. If we write ?= — &lff* (4) i.e. f is the ' equilibrium ' value of the surface elevation, these become I-*—- '£«-?>. S +1 — -4 (t -- B (5) The equation of continuity has the same form as in Art. 193, viz. 3C_ d(hu) aw dt dx dy ' { } whei-e h denotes the depth, from the free surface to the bottom, in the undisturbed condition. This depth will not, of course, be uniform unless the bottom follows the curvature of the free surface as given by (1). 207-208] Plane Sheet of Water 319 If we eliminate £-£from the equations (5), by cross-differentiation, we find 1(1-1)^(1-1)=°' <» or, writing u = d^/dt, v = drjldt, and integrating with respect to t, s-5+-ffi+8— - ;-- (8) This is merely the expression of Helmholtz' theorem that the product of the vorticity 2o> + ^- - y and the cross-section ( 1 + ~- + J- j §# dy, of a vortex-filament, is constant. In the case of a simple-harmonic disturbance, the time-factor being e wt the equations (5) and (6) become iau-2(DV = -g^(£-£), i*v+ 2ayu= -#;-(£- ?), (9) . d(hu) d(hv) m and ^ = -"^ ^T (10) From (9) we find (11) and if we substitute from these in (10), we obtain an equation in f only. In the case of uniform depth the result takes the form v 1 2 r+ <T ^^=v 1 ^ (12) where V 2 2 = d 2 /da? + d 2 /dy 2 , as before. When £=0, the equations (5) and (6) can be satisfied by constant values of u, v, £ provided certain conditions are fulfilled. We must have u--ff, v=£f ,.(13) 2o> dy ' 2a> ex aDd therefore \ [ h > ^ =0 (14) d {x, y) The latter condition shews that the contour-lines of the free surface must be everywhere parallel to the contour-lines of the bottom, but that the value of £ is otherwise arbitrary. The flow of the fluid is everywhere parallel to the contour-lines, and it is therefore further necessary for the possibility of such steady motions that the depth should be uniform along the boundary (supposed to be a vertical wall). When the depth is everywhere the same, the condition (14) is satisfied identically, and the only limitation on the value of f is that it should be constant along the boundary. A simple application of the preceding equations is to the case of free waves in an infinitely long uniform straight canal*. If we assume f =sae iic(ct-x)+my j v = 0, (1) the axis of x being parallel to the length of the canal, the equations (5) of the preceding Art., with the terms in f omitted, give cu = g£, 2(au = -gm£, (2) * Sir W. Thomson, I.e. ante p. 317. 320 Tidal Waves [chap, vm whilst, from the equation of continuity (Art. 207 (6)), c£=hu (3) We thence derive c 2 = gh, m = — 2co/c (4) The former of these results shews that the wave-velocity is unaffected by the rotation. When expressed in real form, the value of £ is f =ae- 2 »yl c cos {k(ct-x) + e} (5) The exponential factor indicates that the wave-height increases as we pass from one side of the canal to the other, being least on the side which is forward in respect of the rotation. If we take account of the directions of motion of a water- particle, at a crest and at a trough, respectively, this result is seen to be in accordance with the tendency pointed out in Art. 202*. It will be observed that there is, in the above solution, no limitation to the breadth of the canal, provided it be uniform. The problem of determining the free oscillations in a rotating canal of finite length, or even the simpler one of reflection of a wave at a transverse barrier, does not however admit of a simple solution by superposition, as was the case in the investigations of Arts. 176, 178. For a wave travelling in the negative direction, we should find f= ft ' 6 2# cos {k (ct + «) + €'}, (6) but this cannot be combined with (5) so as to make u — at a barrier for all values of yf. 209. We take next the case of a circular sheet of water rotating about its centre}. If we introduce polar co-ordinates r, 6, and employ the symbols £, r\ to denote displacements along and perpendicular to the radius vector, then since f = iai;, r) = iar) i the equations (9) of Art. 207 are equivalent to a 2 ?+2 4 W^ = fi r|,(r-r) ) <T 2 ,-2»Wf -g±tf-Q, (1) * For applications to tidal phenomena see Sir W. Thomson, Nature, xix. 154, 571 (1879), and G. I. Taylor, "Tidal Friction in the Irish Sea," Phil. Trans. A, ccxx. 1 (1918). t Poincare\ Legons de Mec. Cel. iii. 124. The problem here indicated has been solved by G. I. Taylor, Proc. Lond. Math. Soc. (2) xx. 148 (1920). He finds that, provided the wave-length (27r/ft) be sufficiently large compared with the breadth (6), there is regular reflection (with a change of phase), in the sense that at a distance from the barrier we have practically superposition of (5) and (6) above, with a' = a, the necessary condition being fc 2 6 2 <7r 2 + 4w 2 & 2 /c 2 . The theory of the free oscillations in a rotating rectangular basin is also discussed in the paper cited. The case where the angular velocity of rotation is relatively small had been previously treated by Eayleigh, Phil. Mag. (6), v. 297 (1903) [Papers, v. 93], and Proc. Roy. Soc. A, lxxxii. 448 (1909) [Papers, v, 497J. J The investigation which follows is a development of some indications given by Kelvin in the paper cited on p. 317. 208-210] Rotating Circular Basin 321 whilsb the equation of continuity (10) becomes d(h&) d(h v ) 6 rdr rdd V } Hence (3) and substituting in (2) we get the differential equation in f. In the case of uniform depth we find (v^+^jr-v^g (4) n2 a 2 i d i a 2 /KX where ^=- 2 + - 8 - + - 2 -- 2 , (5) a 2 -4 and /c 2 = j — (6) gh This might have been written down at once from Art. 207 (12). The condition to be satisfied at the boundary (r = a, say) is f = 0, or ('!-*&«-»-<> < 7 > 210. In the case of the free oscillations we have f=0. The way in which the imaginary i enters into the above equations, taken in conjunction with Fourier's Theorem, suggests that occurs in the form of a factor e is9 , where s is integral. On this supposition, the differential equation (4) becomes a 2 ? 13? / 2 s 2 \ d? + rdr + [ K -r^ = °' (8) and the boundary-condition (7) gives 'g+^=o> < 9 > for r = a. The equation (8) is of Bessel's form, and the solution which is finite for r = may therefore be written ^AJ.Ur)^^^; .. (10) but it is to be noticed that k 2 is not, in the present problem, necessarily positive. When k 2 is negative, we may replace J 8 (/cr) by I s (/cir), where ki is the positive square root of (4&> 2 — cfi)lgK and Z s ( Z 2 2 4 Ia ^ 2 s . 5 ! l 1 + 2(2 5 + 2) + 2.4(2.5 + 2)(2s + 4) + ---j* "'^ U ) In the case of symmetry about the axis (5 = 0), we have, in real form, £= A Jo(tcr). coa (<rt + €), (12) * The functions I a (z) were tabulated by Prof. A. Lodge, Brit. Ass. Rep. 1889. The tables are reprinted by Dale, and by Jahnke and Emde. Extensive tables of the functions e~-*I (z), e~ z I x (z) are given in Watson's treatise. 322 Tidal Waves [chap, viii where k is determined by J ' (*a) = (13) The corresponding values of a are then given by (6). The free surface has, in the various modes, the same forms as in Art. 191, but the frequencies are now greater. If we write c 2 = gh, /3 = 4,co 2 a 2 /c 2 , (14) we have o 2 a 2 \c 2 = K 2 a 2 + /3 (15) It is easily seen, moreover, on reference to (3), that the relative motions of the fluid particles are no longer purely radial ; the particles describe, in fact, ellipses whose major axes are in the direction of the radius vector. For s > we have f =A J 8 (tcr). cos (at + sO + e), (16) where the admissible values of k, and thence of a, are determined by (9), which gives KaJ s '(tca) + — J s (/ea) = (17) G The formula (16) represents a wave rotating relatively to the water with an angular velocity cr/s, the rotation of the wave being in the same direction with that of the water, or the opposite, according as a/a> is negative or positive. If ica is any real or pure imaginary root of (17), the corresponding value of a is given by (15). Some indications as to the values of <r may be gathered from a graphical construction. If we write K 2 a 2 = x, we have, from (6), £=±H)* ™ If we further put *ffi g \ = <t> (k 2 « 2 ), KOJ s {kO) r X the equation (17) may be written 0(^)±fl+f N ) =0 (19) The curve #=-<£(#) (20) can be readily traced by means of the tables of the functions J 8 (z), I 8 (z) ; and its inter- sections with the parabola y 2 = l+^/3 (21) will give, by their ordinates, the values of ct-/2co. The constant /3, on which the positions of the roots depend, is equal to the square of the ratio 2coa/(gh)^ which the period of a wave travelling round a circular canal of depth h and perimeter 2ira bears to the half- period (jt/o>) of the rotation of the water. The diagrams on the next page indicate the relative magnitudes of the lower roots, in the cases s=l and s = 2, when /3 has the values 2, 6, 40, respectively*. * For clearness the scale of y has been taken to be 10 times that of x. 210] Free Oscillations 323 With the help of these figures we can trace, in a general way, the changes in the character of the free modes as /3 increases from zero. The results may be interpreted as due either to a continuous increase of co, or to a continuous diminution of h. We will use [•-«] 324 Tidal Waves [chap. VIII the terms 'positive' and 'negative' to distinguish waves which travel, relatively to the water, in the same direction as the rotation and the opposite. When /3 is infinitely small, the values of x are given by J 8 ' (#z)=0; these correspond to the vertical asymptotes of the curve (20). The values of cr then occur in pairs of equal and oppositely- signed quantities, indicating that there is now no difference between the velocity of positive and negative waves. The case is, in fact, that of Art. 191 (12). As /3 increases, the two values of a- forming a pair become unequal in magnitude, and the corresponding values of x separate, that being the greater for which o-/2a> is positive. When /3=s (s + 1) the curve (20) and the parabola (21) touch at the point (0, -1), the corresponding value of o- being — 2a>. As /3 increases beyond this critical value, one value of x becomes negative, and the corresponding (negative) value of o-/2a> becomes smaller and smaller. Hence, as /3 increases from zero, the relative angular velocity becomes greater for a negative than for a positive wave of (approximately) the same type ; moreover the value of a for a negative wave is always greater than 2w. As the rotation increases, the two kinds of wave become more and more distinct in character as well as in ' speed.' With a sufficiently great value of /3 we may have one, but never more than one, positive wave for which cr is numerically less than 2o>. Finally, when /3 is very great, the value of o- corre- sponding to this wave becomes very small compared with 2o>, whilst the remaining values tend all to become more and more nearly equal to + 2o. If we use a zero suffix to distinguish the case of w = 0, we find o- 2 _ K 2 + 4co 2 /gh _ x + p .(22) where x refers to the proper asymptote of the curve (20). This gives the 'speed' of any free mode in terms of that of the corresponding mode when there is no rotation. The preceding statements are illustrated by the following table, which gives for the case of s = 1 approximate values of <a within the range of the upper diagram on p. 323, together with the corresponding values of o-/2o> and aa/c. (3 = ,8=2 ,3=6 j8 = 40 j8 = co Ka = aa/c Ka <r/2w a-a/c Ka <r/2o> <xa/c Ka o-/2w <rajc Ka (r/2w <ra\e ±1-84 ±533 J2-19 I o (5-38 (5-28 + 1-84 -1-00 + 3-93 -3-86 + 2-61 -1-41 + 5-56 -5-47 J2-29 \2-10i p-41 (5-25 + 1-37 -0-51 + 2-42 -2-37 + 3-35 -1-26 + 5-94 -5-79 J2-38 [6'2Si J5-47 \5-18 + 1-07 -0-17 + 1-32 -1-29 + 6-76 -1-09 + 8-36 -8-17 T2-40 J5-52 (5-14 + 1-00 + 1-00 -1-00 -l'OO +0* -0* 211. As a sufficient example of forced oscillations we may assume i-O ,i(<rt + s9+e) .(23) where the value of a is now prescribed. This makes V x 2 f = 0, and the equation (4) then gives f=4/ s (*r)^+^ +e ), .(24) 2io-2ii] Forced Oscillations 325 where A is to be determined by the boundary-condition (7), viz. 2^ s 1 + i") ^= i — ^ .a (25) T . , . , 2«ft) r / \ /eaJ g («a) H J s (/ea) a* This becomes very great when the frequency of the disturbance is nearly coincident with that of a free mode of corresponding type*. From the point of view of tidal theory the most interesting cases are those of s = 1 with o- = g), and s = 2 with <r=2o), respectively. These would represent the diurnal and semidiurnal tides due to a distant disturbing body whose proper motion may be neglected in comparison with the rotation <o. In the case of s = 1 we have a uniform horizontal disturbing force. Putting, in addition, <r = (o, we find without difficulty that the amplitude of the tide-elevation at the edge (r = a) of the basin has to its 'equilibrium-value' the ratio /i(i)+i4(i)' K } where 2=^(30). With the help of Lodge's tables we find that this ratio has the values 1-000, -638, -396, for /3= 0, 12, 48, respectively. When <r = 2o), we have k=0, and thence, by (23), (24), (25), C = l (27) i.e. the tidal elevation has exactly the equilibrium-value. This remarkable result can be obtained in a more general manner ; it holds whenever the disturbing force is of the type J =x ( r ) c i(2«t + tf + e) ( 28 ) provided the depth h be a function of r only. If we revert to the equations (1), we notice that when o- = 2a> they are satisfied by £= £, r) = i%. To determine £ as a function of r, we substitute in the equation of continuity (2), which gives f-^=-xW (29) The arbitrary constant which appears on integration of this equation is to be determined by the boundary-condition. In the present case we have x(r) = 8 / a *- Integrating, and making £ = for r = a, we find Or 8 ' 1 A£=~T- ( a 2-r 2 )e*( 2 ^ + 8fl+ <) (30) The relation r) = ig shews that the amplitudes of £ and r\ are equal, while their phases differ by 90° ; the relative orbits of the fluid particles are in fact circles of radii Cr*- 1 r= u^^-^ < 31 > described each about its centre with angular velocity 2a> in the negative direction. We may easily deduce that the path of any particle in space is an ellipse of semi-axes r±r described about the origin with harmonic motion in the positive direction, the period being 27r/©. This accounts for the peculiar features of the case. For if £ have always the * The case of a nearly circular sheet is treated by Proudman, "On some Cases of Tidal Motion on Eotating Sheets of Water," Proc. Lond. Math. Soc. (2) xii. 453 (1913). 326 Tidal Waves [chap, viii equilibrium-value, the horizontal forces due to the elevation exactly balance the disturbing force, and there remain only the forces due to the undisturbed form of the free surface (Art. 207 (1)). These give an acceleration gdzjdr, or o>V, to the centre, where r is the radius vector of the particle in its actual position. Hence all the conditions of the problem are satisfied by elliptic-harmonic motion of the individual particles provided the positions, the dimensions, and the 'epochs' of the orbits can be adjusted so as to satisfy the con- dition of continuity, with the assumed value of £. The investigation just given resolves this point. When the sheet of water is bounded also by radial walls the problem is more difficult. The tidal oscillations (free and forced) in a semicircular basin of uniform depth are discussed by Proudman*, with an application to the tides of the Black Sea, the disturbing forces being of the idealized diurnal and semi-diurnal types. The free and forced oscillations in a rotating elliptic basin of uniform depth are discussed by Goldstein f. 212}. We may also notice the case of a circular basin of variable depth, the law of depth being the same as in Art. 193, viz. h = Ao ( 1 -^ « Assuming that £, rj, £ all vary as e i ( <rt + «* +€ ), and that A is a function of r only, we find, from Art. 209 (2), (3), --v>c + .g(j + Stt-a + *(S + i4-aK-ft-o m (i-S)0+S-SO-5(^ + ^) + -- f - ( Introducing the value of h from (1), we have, for the free oscillations, o- 2 -4o) 2 9h This is identical with Art. 193 (6), except that we now have o- 2 — 4co 2 4cos gk va 2 in place of (r 2 jgk . The solution can therefore be written down from the results of that Art., viz. if we put (<r 2 -4co 2 )a 2 4cos , . „ — PT — = n{n-Z)-s\ (4) we have f=4,QVL^7 ) ^e i ( <rt + 8( ' + e ), (5) where a=%n+§s, /3=l+iU-|rc, y = s + l; and the condition of convergence at the boundary r = a requires that n = s + 2j, (6) where,;' is some positive integer. The values of a are then given by (4). The forms of the free surface are therefore the same as in the case of no rotation, but the motion of the water particles is different. The relative orbits are in fact now ellipses having their principal axes along and perpendicular to the radius vector; this follows easily from Art. 209 (3). * M. N. R. A. S., Geophys. Suppt. ii. 32 (1928). t Ibid. ii. 213 (1929). X See the footnote to Art. 193. 2ii-2i2] Basin of Variable Depth 327 In the symmetrical modes (* = 0), the equation (4) gives 0-2 = ^2 + 4^ ( 7 ) where <r denotes the ' speed ' of the corresponding mode in the case of no rotation, as found in Art. 193. For any value of s other than zero, the most important modes are those for which n = s + 2. The equation (4) is then divisible by o- + 2o>, but this is an extraneous factor ; discarding it, we have the quadratic o- 2 -2o>o-=2s^°, (8) a 1 whence o-=<*± L*+2s 9 -^y . (9) This gives two waves rotating round the origin, the relative wave-velocity being greater for the negative than for the positive wave, as in the case of uniform depth (Art. 210). With the help of (8) the formulae reduce to <-*©'• e-*^©'"'. " =ii i A -&" 1 ' (10) the factor e l i <rt + s6 + e ) being understood in each case. Since 17— *£, the relative orbits are all circles. The case 5=1, n = S, is noteworthy; the free surface is then always plane, and the circular orbits have all the same radius. In the following table, which relates to this case, /3 stands for 4o 2 a 2 /c 2 , where c = *J(gh ). p=o ,3=2 £=6 /3 = 40 aajco tr/2w <rajc <r/2w <ra/c <r/2w <ra/c ±1-414 + 1-618 -0-618 + 2-288 -0-874 + 1-264 -0-264 + 3-096 - 0-646 + 1 -048 -0-048 + 6-626 -0-302 When ?i>5 + 2, we have nodal circles. The equation (4) is then a cubic in o-/2to; it is easily seen that its roots are all real, lying between - co and -1,-1 and 0, and +1 and + oo , respectively. The following table is calculated for the case of s = 1, n = 5. ,8=0 i P = 2 13 = 6 I /3 = 40 <xa/c <r/2w cra/co <r/2w aajc cr/2w <ralc ±3-742 + 2-889 -0-125 -2-764 + 4-085 -0-176 -3-909 + 1-874 -0-100 -1-774 + 4-590 -0-245 -4-344 + 1-183 - 0-040 -1-143 + 7-483 -0-253 -7-230 The first and the last root of each triad give positive and negative waves of a somewhat similar character to those already obtained in the case of uniform depth. The smaller negative root gives a comparative slow oscillation which, when the angular velocity o> is infinitely small, becomes a steady rotational motion, without elevation or depression of the surface. The possibility of oscillations of this type was pointed out in Art. 206, ad fin. In 328 Tidal Waves [chap, viii the present case the transition is easily traced. It follows from (4) that the relevant limiting value of o-/2co, when co is infinitesimal, is - } . We then find, from Art. 209 (2), (3), ^cfl-^jeHO + 't), ^wU-bfSeW*^ (11) with £= -*^r (l - f -f) e i( ' + <r<) , (12) ultimately, where <r=— fco. The most important type of forced oscillations is such that l=C (-Ye^+80 + e) (13) We readily verify, on substitution in (3), that *" 2sgh -(v*-2a>o-)a 2 ^ ^ We notice that when a = 2a> the tide-height has exactly the equilibrium-value, in agree- ment with Art. 211. If <r 1? o- 2 denote the two roots of (8), the last formula may be written ^(l-er/erxXl-T/^) (15) The tidal oscillations in a semicircular basin with the above law of depth have been examined by Goldsbrough *. The difficulty of the problem consists in satisfying the conditions at the straight portion of the boundary. 212 a. Place may be found here for one or two illustrations of the approxi- mate procedure outlined in Art. 205 a. 1°. To take first a known problem, that of the circular basin of uniform depth (Art. 210). Assuming as the polar co-ordinates of a displaced particle, relative to an initial line revolving with the angular velocity o>, r'=r+& 6' = 0+T)/r, (1) the equation of continuity is h dr r rd0 ' as in Art. 209 (2). With our previous notation •(2) (3) /a f2n fa f2ir "\ J H'+Wrdddr, r-T =igpJJ o ?rd6dr, M> = phj a J^{^-4)rdedr. We take as our assumed type, for the gravest mode, $ = A (l - ^ cos (*t + 6), V = (-A + B r ^ sin(o-< + 0), (4) which make jr=(3A-B)^cos{<rt + 0) (5) The constants in (4) have been adjusted so that £ shall be finite for r — 0. * Proc. Roy. Soc. cxxii. 228 (1929). 212-212 a] Approximate Method 329 Hence with the definitions of Art. 205 a, taking the mean values of the functions in (3), and performing the integrations, P=^7rpha 2 (4:A 2 -SAB+B 2 ) } Q= -lirpcohcfi^A 2 - AB), R=%7rgph 2 (3A-B) 2 . ...(6) If we write for shortness c=s/feA)i <ra/c = x, 4a> 2 a 2 /c 2 = 0, (7) the equation (r 2 P+aQ-R = (8) becomes (4x 2 -3j(3x-\ 7 -)A 2 -(Sx 2 - s /(3x-9)AB + (x 2 -%)B 2 = (9) The stationary values of uc are then given by x 2 (7x 2 - 6Vj8a?-/3- 24) = (10) The zero roots may be disregarded as corresponding to a merely circulatory motion, without change of surface-level. To compare with the numerical results of Art. 210 (p. 324) we put /3 = 2, 6, 40 in succession. The finite roots of (10) are -1-43) -1-271 + 2-65J' +3-27] ' -1-35) + 6-77 in the respective cases. It is only in the third case that there is any serious deviation from the correct value. It will be seen that the approximate method is fairly successful over a considerable range of the parameter /3. 2°. In the case of a rectangular basin of uniform depth, we take axes Ox, Oy coincident with two of the sides, whose lengths are (say) a, b, respectively. Denoting by £, 77 the com- ponent displacements of a particle, we have Z=WJ a J\p+v*)<tedy, V-T»=\9P j'J^dxdy^ Let us assume as an approximate type / (&l-rih)dxdy. .(11) £ == A sin — cos a-t, rj = B sin -~ sin at (12) a This is suggested by the case of o> = 0, where either A or B is zero, and cannot be expected to give a good result for more than a limited range of a>. From (12) we derive C d£ dn (A ttx B try . \ ,„. y = - ~ -7T = — w — cos — cos o-*+ t-cos-/- sin at) (13) h dx dy \a a b J v ' Hence P=\phab(A*+&\ qJ^^AB, R=\* 2 gph 2 (±A 2 + \B^ (14) The equation (8) now takes the form (<r>-^)A* + ?^AB + ^-^)B> = 0, (15) where c 2 =gh as before. The stationary values of a are therefore given by (^-<^W)= 2 -^-\ (i6) where <r u a 2 are the values of a corresponding to oscillations parallel to x and y, respec- tively, when there is no rotation. 330 Tidal Waves [chap, viii If a) is small and a, b decidedly unequal, then in the type where <r = o- x , nearly, we have 128c (T- 0"i= -47- o 2\) \ 1 ' ) 7r 4 (cr^-(r2 2 ) approximately. The corresponding ratio BjA is then given by 7T and is accordingly small, as was to be expected. For a square tank (a = b), on the other hand, (16) makes ISapA+W-flB-O, (18) . 2 _ . 1 2 =± — , 9) '-n-±% (*» approximately. Then B/A = + 1 . Tides on a Rotating Globe. 213. We proceed to give some account of Laplace's problem of the tidal oscillations of an ocean of (comparatively) small depth covering a rotating globe *. In order to bring out more clearly the nature of the approximations which are made on various grounds, we adopt a method of establishing the fundamental equations somewhat different from that usually followed. When in relative equilibrium, the free surface is of course a level-surface with respect to gravity and centrifugal force ; we shall assume it to be a surface of revolution about the polar axis, but the ellipticity will not in the first instance be taken to be small. We adopt this equilibrium-form of the free surface as a surface of reference, and denote by and <j> the co-latitude (i.e. the angle which the normal makes with the polar axis) and the longitude, respectively, of any point upon it. We shall further denote by z the altitude, measured outwards along a normal, of any point above this surface. The relative position of any particle of the fluid being specified by the three orthogonal co-ordinates 0, <f>, z, the kinetic energy of unit mass is given by 27 7 =(i2 + *) 2 <9 2 +GT 2 (a> + <£) 2 -fi 2 (1) where R is the radius of curvature of the meridian-section of the surface of reference, and ta is the distance of the particle from the polar axis. It is to be noticed that R is a function of only, while -or is a function of both and z ; and it easily follows from geometrical considerations that = cos 0, «- = sin (2) (R + z)d0 ' dz * "Recherches sur quelques points du systeme du monde," Mem. de V Acad. roy. des Sciences, 1775 [1778] and 1776 [1779]; Oeuvres Completes, ix. 88, 187. The investigation is reproduced, with various modifications, in the Mecanique Ctleste, Livre 4 me , c. i. (1799). 212 a-213] Laplace's Theory of the Tides 331 The component accelerations are obtained at once from (1) by Lagrange's formula. Omitting terms of the second order, on account of the restriction to infinitely small motions, we have R+z\dtcQ ddj v R (co 2 + 2 G >4>)a>~ i{jt d ^- d i)=^ +2a} { fc ■)■ (3) ^8i-S = '- (< ° 2+2 ^ )CT ^ =- — 2&W COS Oh dv + 2am cos 6 + 2&>w sin (5) Hence, if we write a, v, w for the component relative velocities of a particle, viz. U = (R+ z)0, V = 'BT<j), w=z } (4) and make use of (2), the hydrodynamical equations may be put in the forms 1 d_ dt t»Td<t)\p — — 2&>v sin = — dt oz \p where M* is the gravitation-potential due to the earth's attraction, whilst H denotes the potential of the disturbing forces. So far the only approximation has consisted in the omission of terms of the second order in u, v, w. In the present application, the depth of the sea being small compared with the dimensions of the globe, we may replace R + z by R. We will further assume that the vertical velocity w is small compared with the horizontal components u, v and that dw/dt may be neglected in comparison with cov. As in the theory of 'long' waves, such assumptions are justified a posteriori if the results obtained are found to be consistent with them (cf. Art. 172)*. Let us integrate the third of equations (5) between the limits z and f, where f denotes the elevation of the disturbed surface above the surface of reference. At the surface of reference (z = 0) we have "9 — ^co 2 ™ 2 = const., by hypothesis, and therefore at the free surface (z = f ) *& — \cd 2 ^ 2 = const. + #£ 8 approximately, provided g = dz (¥- ico 2 ^ 2 ) z=0 •(6) * Thus in the simplified conditions of Arts. 219, 220 wjwv is of the order m( = «%/#). 332 Tidal Waves [chap, viii Here g denotes the value of apparent gravity at the surface of reference ; it is of course, in general, a function of 6, but its variation with z is neglected. The integration in question then gives ^ + ^-ift> 2 OT 2 = const. +#?+2G>sin0 | vdz, (7) p Jz where the variation of the disturbing potential CI with z has been neglected in comparison with g. The last term is of the order of a>hv sin 6, where h is the depth of the fluid, and it may be shewn that in the subsequent applications this is of the order h/a as compared with g£ *. Hence, substituting in the first two of equations (5), we obtain, with the approximations indicated, where f =~n/# (9) These equations are independent of z, so that the horizontal motion may be assumed to be sensibly the same for all particles in the same vertical line. As in Art. 198, this last result greatly simplifies the equation of continuity. In the present case we find without difficulty dj_ 1 \d(hwu) d(hv)] dt~ v\ Rdd + d<t> j u; It is important to notice that the preceding equations involve no assumptions beyond those expressly laid down; in particular, there is no restriction as to the ellipticity of the meridian, which may be of any degree of oblateness. 214. In order, however, to simplify the question as far as possible, with- out sacrificing any of its essential features, we now take advantage of the circumstance that in the actual case of the earth the ellipticity is a small quantity, being in fact comparable with the ratio (co 2 a/g) of centrifugal force to gravity at the equator, which ratio is known to be about ^J^. Subject to an error of this order of magnitude, we may put R = a, sr — a sin 6, g — const., where a is the earth's mean radius. We thus obtain ^_2™cos0 = -^(f-f), |%2om cos = - 2 -JL-; (£-?), dt a oa s " dt asm6d<j> . „ (1) .,, 3? 1 (d(kuBind) , d(hv)) with £ = ; y ' + _A^l (2) dt a sin 6 { dO dcf> J v ; this last equation being identical with Art. 198 (l)f- * This, again, may be verified in the same cases. The upshot is that the vertical acceleration is neglected, as in the theory of 'long' waves. f Except for the notation these are the equations arrived at by Laplace, I.e. ante p. 330. 213-215] Fundamental Equations 333 Some conclusions of interest follow at once from the mere form of the equations (1). In the first place, if u, v denote the velocities along and perpendicular to any horizontal direction s, we easily find, by transformation of co-ordinates, g_2»vooefl — y|(?-f) (3) In the case of a narrow canal, the transverse velocity v is zero, and the equation (3) takes the same form as in the case of no rotation ; this has been assumed by anticipation in Art. 183. The only effect ot the rotation in such cases is to produce a slight slope of the wave-crests and furrows in the direction across the canal, as investigated in Art. 208. In the general case, resolving at right angles to the direction of the relative velocity (q, say), we see that a fluid particle has an apparent acceleration 2coq cos 6 towards the right of its path, in addition to that due to the forces. Again, by comparison of (1) with Art. 207 (5), we see that the oscillations of a sheet of water of relatively small dimensions, in co-latitude 6, will take place according to the same laws as those of a plane sheet rotating about a normal to its plane with angular velocity co cos 0. As in Art. 207, free steady motions are possible, subject to certain conditions. Putting f=0, we find that the equations (1) and (2) are satisfied by constant values of u, v, f, provided u ■ J. K v = $_ d I (4) 2coa sin 6 cos d<f> ' 2coacos0d0' v 7 dJ mw^ < 5 > The latter condition is satisfied by any assumption of the form ?=/(Asec0), (6) and the equations (4) then give the values of n, v. It appears from (4) that the velocity in these steady motions is everywhere parallel to the contour- lines of the disturbed surface. If h is constant, or a function of the latitude only, the only condition imposed on f is that it should be independent of <£ ; in other words the elevation must be symmetrical about the polar axis. 215. We shall suppose henceforward that the depth h is a function of only, and that the barriers to the sea, if any, coincide with parallels of latitude. We take first the cases where the disturbed form of the water-surface is one of revolution about polar axis. When the terms involving <f> 334 Tidal Waves [chap, viii are omitted, the equations (1) and (2) of the preceding Art. take the forms ™2ewcos0 = -2^(£ -■£)", ^ + 2cou cos = 0, (1) d£ d(hu sm0) With 37 = r— — -j- (2) dt a sin Odd v Assuming a time-factor e i<Tt , and solving for m, v, we find _ jgg ?/f_p> = _ 2&)(7 cos d ,j, p. /ox M ~o- a -4ft> 2 cos 2 <9a90 U gj ' ? o- 2 - 4 a> 2 cos 2 (9 a 90^ Q > '" {) .,, . o 9 (Aw sin 0) ,., with i(7?= ■ a-^ (4) a sin Odd The formulae for the component displacements (f, 77, say) can be written down from the relations u = j, = ^ r or w = icrf , = 1*770-. It appears that the fluid particles describe ellipses having their principal axes along the meridians and the parallels of latitude, respectively, the ratio of the axes being a/2co . sec 6. In the forced oscillations of the present type the ratio cr/2o) is very small ; so that the ellipses are very elongated, with the greatest length from E. to W., except in the neighbourhood of the equator. Eliminating u and v between (3) and (4), and writing, for shortness, <-*-r> £-■* T= m ' (5) -** ^(/^yiw=- 4 <- < 6 > In the case of uniform depth, this becomt k§&%)+ K ~-* : (7) where a = cos 0, and 3=—— == — — (8) h gh 216. First, as regards the free oscillations. Putting £=0, we have *&$$+"-*> < 9 > and we notice that in the case of no rotation this is included in (1) of Art. 199, as may be seen by putting fif 2 = a 2 a 2 /gh, f= 00 . The general solution of (9) is necessarily of the form K=AF(j*) + Bf{ii). (10) where F(/jl) is an even, and f(p) an odd, function of fju, and the constants A, B are arbitrary. In the case of a zonal sea bounded by two parallels of latitude, the ratio A : B and the admissible values of / (and thence of the frequency cr/27r) are determined by the conditions that u = at each of these parallels. If the boundaries are symmetrically situated on opposite sides 215-216] Case of Symmetry 335 of the equator, the oscillations fall into two classes; viz. in one of these B = Q, and in the other A = 0. By supposing the boundaries to contract to points at the poles, we pass to the case of an unlimited ocean, and the admissible values of f are now determined by the condition that n must vanish for /x = ± 1. The argument is, in principle, exactly that of Art. 201, but the application of the last-mentioned condition is now more difficult, owing to the less familiar form in which the solution of the differential equation is obtained. In the case of symmetry with respect to the equator, we assume, following the method of Kelvin* and Darwin f, — j-r a d ^ = B lf , + B^+... + B 2j+1 ^^ + (11) This leads to f = .4 - ifBvf + i {Bi -f*B 3 ) p* + . . . + 1 (B+, -/» Vi) /** + •••. • • -(12) where A is arbitrary ; and makes ^(^j^)-B 1 + S(B d -B 1 )^ + ...+(2j + l)(B ij+1 -B ij _ 1 )^+.... (13) Substituting in (9), and equating coefficients of the several powers of /j,, we find Bt-pA-O, (14) M'rfS*-* < 15 > and thenceforward B *"-{ 1 -y(ij+l)) B +*^3j(»j + l) B *-*- (16) These equations determine B\,Bz, ... B 2 j+i, ... in succession, in terms of A,a,nd the solution thus obtained would be appropriate, as already explained, to the case of a zonal sea bounded by two parallels in equal N. and S. latitudes. In the case of an ocean covering the globe, it would, as we shall prove, give infinite velocities at the poles, except for certain definite values of/. Let us write B 2 j + i/B 2 j-i — Nj+i ; (17) we shall shew, in the first place, that as j increases Nj must tend either to the limit or to the limit 1. The equation (16) may be written at i /3/ 2 | g 1 ns> iV ^ 1 - 1 2j(2j + l) + 2j(2j + l)N j (l *> * Sir W. Thomson, "Note on the 'Oscillations of the First Species ' in Laplace's Theory of the Tides," Phil. Mag. (4), 1. 279 (1875) [Papers, iv. 248]. f " On the Dynamical Theory of the Tides of Long Period," Proc. Roy. Soc. xli. 337 (1886) [Papers, i. 336]. 336 Tidal Waves [chap, viii Hence, whenj is large, either ^=W)' ; (19) approximately, or N j+1 is not small, in which case Nj+% will be nearly equal to 1, and the values of iV^+s, Nj+i, ... will tend more and more nearly to 1, the approximate formula being _ _ ^ =1 -§w^ (20) Hence, with increasing j, N t tends to one or other of the forms (19) and (20). In the former case (19), the series (11) will be convergent for ja — ± 1, and the solution will be valid over the whole globe. In the other event (20), the product N s iV 4 . . . N j+J , and therefore the coefficient i?2; + i, tends with increasing j to a finite limit other than zero. The series (11) will then, after some finite number of terms, become com- parable with 1 + (A 2 + fjL l + ..., or (1 -fi 2 )- 1 , so that we may write 1 £-£ + t^'. (21) where L and M are functions of /x which remain finite when \x = ± 1. Hence, from (3), "■~ ^^f^-kv-^^-^M), -.(22) which makes u infinite at the poles. It follows that the conditions of our problem can be satisfied only if N t tends to the limit zero ; and this consideration, as we shall see, restricts us to a determinate series of values of/ 2j(2;+l) ^^^ . (23) 2j(2j + l)-^ and by successive applications of this we obtain N s in the form of a convergent continued fraction £ /3 y 2j(2j + l) (2j+2)(2? + 3) (2? + 4)(2j + 5) ' 1 W* .I tif % .I £/ 2 , ' 2j(2j + l) +1 (2j + 2)(2j + 3) +i (2j + 4)(2j + 5) + - (24) on the present supposition that N j+t tends with increasing k to the limit 0, in the manner indicated by (19). In particular, this formula (24) determines the value of N t . Now from (15) we must have *i-l-§?S. < 25 > 216] Free Oscillations 337 when ce 1 -|L- + -— ^- — gf— ~ °> (26) 4.5 + 6.7 + "' which is equivalent to jVj = oo . This equation determines the admissible values of/(= <t/2g>). The constants in (11) are then given by B 1 = j3A t B z =JST 2 pA, B^NzNzpA,..., (27> where A is arbitrary. It is easily seen that when j3 is infinitesimal the roots of (26) are given by a *£=V/*=n(n + l), (28) where n is an even integer; cf. Art. 199. One arithmetically remarkable point remains to be noticed. It might appear at first sight that when a value of/ has been found from (26) the coefficients B 3 , B 5 , B 7 , ... could be found in succession from (15) and (16), or by means of the equivalent formula (18). But this would require us to start with exactly the right value of / and to observe absolute accurac}' in the subsequent stages of the work. The above argument shews, in fact, that any other value, differing by however little, if adopted as a starting point for the calculation will inevitably lead at length to values of N, which approximate to the limit 1 *. An approximation to the longest free period may be attempted by the method of Art. 205 a. Denoting by £, r] the displacements southwards and eastwards, respectively, we have, in the notation of the Art. referred to, irpha 2 f" (i 2 + i7 2 ) sin 8d8, M'=2ir P ha 2 (* (£ 2 + i] 2 ) sin 8 d8, M' = 2irpha 2 \ (gcos 8. rj -rj . £cos 6) sin 8d8, V- T =7rgpa 2 f * £ 2 sin 8d8. ...(29) We will assume that as in the case of no rotation the surface elevation is represented by a zonal harmonic of the second order. The formulae (3) of Art. 215 then suggests for our assumed type £ = A sin 8 cos 8 cos at, rj — B sin 8 cos 3 8 sin at, (30) which makes C= ?-^(£sin0)=--(3cos 2 0-l),icoso-* (31) a sin 8 d8 ^ ' a x ' ' We find P= 7rp ka 2 (^A 2 + ^ E B i ), Q=&,ir pa >ha*AB, R = ±7rgph 2 A 2 (32) The equation (10) of Art. 205 a becomes (x 2 -6) A 2 + J(3. %xAB + %B 2 x 2 =0, (33) where x=aal s f{gh\ p = 4o} 2 a 2 /gh ...(34) The stationary values of x are then given by #2 = 6 + 10 (35) * Sir W. Thomson, I.e. ante p. 335. 1 338 Tidal Waves [chap, viit For example, taking /3 = 5, which would correspond in the case of the earth to a depth of 58080 ft., we find aa/J(gh) = 2"854, co/a = '3917. The latter number gives the period in terms of the sidereal day. Hence in sidereal time 27r/o- = 9h. 24 m. The true period, as calculated by Hough (see Art. 222) is 9h. 52 m., but this allows for the mutual gravitation of the disturbed water, which we have neglected. A correction is however easily made. Since we neglect effect of centrifugal force on gravity the influence of T in (29) may be disregarded, whilst the value of V is altered in the ratio l-f^ = '892, Po where p 1 /p ( = '18) is the ratio of the density of the water to the mean density of the earth (see Art. 200). The result is to replace (35) by # 2 =5'352 + f/3 ^36) For /3 = 5 this gives a period oi y n. 48 m., in close approximation to Hough's value. For greater values of /3, i.e. smaller depths of the ocean, or greater speeds of rotation, the approximation is less satisfactory, as we should expect from the nature of our assumed type. 217. It is shewn in the Appendix to this Chapter that the tide-generating potential, when expanded in simple- harmonic functions of the time, consists of terms of three distinct types. The first type is such that the equilibrium tide-height would be given by £=#'(i-cos 3 0).cos(a-*-fe) (37)* The corresponding forced waves are called by Laplace the ' Oscillations of the First Species ' ; the}' include the lunar fortnightly and the solar semi-annual tides, and generally all the tides of long period. Their characteristic is symmetry about the polar axis, and they form accordingly the most important case of forced oscillations of the present type. If we substitute from (37) in (7), and assume for 2 ,» ^- and £ expressions of the forms (11) and (12), we have, in place of (14), (15), B^WH'-PA-O, (38) ^-(l-f^i + i/S^O, (39) whilst (16) and its consequences hold for all the higher coefficients. It may be noticed that (39) may be included under the general formula (16), provided we write B_x = — 2H'. It appears by the same argument as before that the only admissible solution for an ocean covering the globe is the one that makes N^ - 0, and that accordingly Nj must have the value given by the continued fraction in (24), where / is now prescribed by the frequency of the disturbing forces. * In strictness, here denotes the geocentric latitude, but the difference between this and the geographical latitude may be neglected consistently with the assumptions introduced in Art. 214. 216-217] Tides of Long Period 339 In particular, this formula determines the value of N-y . Now B 1 = N 1 B_ l = -2N 1 H', and the equation (38) then gives A = -W-^N l H'; (40) in other words, this is the only value of A which is consistent with a zero limit of N 5 , and therefore with a finite velocity at the poles. Any other value of A, if adopted as a starting point for the calculation of B\, B 3 , B 5 , ... in succession, by means of (38), (39), and (16), would lead ultimately to values of Nj approximating to the limit 1. Moreover, since absolute accuracy in the initial choice of A and in the subsequent computations would be essential to avoid this, the only practical method of calculating the coefficients is to use the formulae B X \W = -2N x , B 3 = N % B lt B 5 = N S B 3) or Bt/H' = - 2 A" a , B s /H' = - 2N X N 2 , B 5 /H' = - 2^ AWs, • . . (41) where the values of A 7 i, N%, N 3 , ... are to be computed from the continued fraction (24). It is evident a posteriori that the solution thus obtained will satisfy all the conditions of the problem, and that the series (12) will converge with great rapidity. The most convenient plan of conducting the calculation is to assume a roughly approximate value, suggested by (19), for one of the ratios Nj of sufficiently high order, and thence to compute N t . lt Jf/-i, ... N*, tfi in succession by means of the formula (23). The values of the constants A, B 1} Z? 3 , ..., in (12), are then given by (40) and (41). For the tidal elevation we find m' = - 2AV/9 - (1 -f*N y ) ^ - JJTi (1 -PNz) ^-... -\SiNi...N, ,(l.-/»ffi)Ai*- (42) In the case of the lunar fortnightly tide, / is the ratio of a sidereal day to a lunar month, and is therefore equal to about ^V> or more precisely '0365. This makes f 2 = '00133. It is evident that a fairly accurate representation of this tide, and a fortiori of the solar semi-annual tide, and of the remaining tides of long period, will be obtained by putting /= ; this materially shortens the calculations. The results will involve the value of /3, =4f(o 2 a 2 /gh. For £ = 40, which corresponds to a depth of 7260 feet, we find in this way £/#' = -1515 - 1-0000//2+ 1-5153/x 4 - T2120/* 6 + -6063/* 8 - -2076m 10 + -0516/i 12 - -0097/u 14 '+ '0018/* 16 - '0002^ 18 , (43)* * The coefficients in (43) and (44) differ only slightly from the numerical values obtained by Darwin for the case f= -0365. 340 Tidal Waves [chap, viii whence, at the poles (//,= ± 1), ?=-f^'x-154, and, at the equator (/j, = 0), ?= Jtf'x'455. Again, for /3 = 10, or a depth of 29040 feet, we get QH' = -2359 - 1 -0000^ 2 + -5898/i 4 - '1623/* 6 + -0258^ - -0026/x 10 + -0002^ 12 (44) This makes, at the poles, ?=-§#' x-470, and, at the equator, f = J£T x -708. For /3 = 5, or a depth of 58080 feet, we find ?/£T = -2723 - 1-0000/x 2 + 3404^ 4 - -0509/x 6 + -0043/t 8 - 0004ya 10 (45) This gives, at the poles, and, at the equator, f = \H' x -817. Since the polar and equatorial values of the equilibrium tide are — f #' and \R' , respectively, these results shew that for the depths in question the long-period tides are, on the whole, direct^ though the nodal circles will, of course, be shifted more or less from the positions assigned by the equi- librium theory. It appears, moreover, that, for depths comparable with the actual depth of the sea, the tide has less than half the equilibrium value. It is easily seen from the form of equation (7) that with increasing depth, and consequent diminution of /3, the tide-height will approximate more and more closely to the equilibrium value. This tendency is illustrated by the above numerical results. It is to be remarked that the kinetic theory of the long-period tides was passed over by Laplace, under the impression that practically, owing to the operation of dissipative forces, they would have the values given by the equilibrium theory. He proved, indeed, that the tendency of frictional forces must be in this direction, but it has been maintained by Darwin* that in the case of the fortnightly tide, at all events, it is doubtful whether the effect would be nearly so great as Laplace supposed. We shall return to this point later. 218. When the disturbance is no longer restricted to be symmetrical about the polar axis, we must recur to the general equations (1) and (2) of Art. 214. We retain, however, the assumptions as to the law of depth and the nature of the boundaries introduced in Art. 215. * I.e. ante p. 335. 217-219] Diurnal Tides 341 If we assume that O, u, v, fall vary as e l ' (<rt+ ** +<) , where s is integral, th equations referred to give i*u-2tovceQ0=-2-3 h (Z-'E) i iav + 2coucosd=- -^L(f-{), ...(1) ad6 x asmO ^ ; v ' ... 1 (3 (Aw sin 0) . . ) with lo-? = ^-^ l-^— 5Z + W M ( 2 ) a sin ( 90 J v ' Solving for u, v, we find a /cos 3f , . 4m (/ 2 - cos 2 6)\ f dd b .(3) where we have written G) 2 tt ?-?=£' a-/- y-» W as before. It appears that in all cases of simple-harmonic oscillation the fluid particles describe ellipses having their principal axes along the meridians and parallels of latitude, respectively. Substituting from (3) in (2) we obtain the differential equation in f ' : _3 f Asin0 (d£ ' s_ , \] sin (980 !/ 2 -cos 2 0\a0 + / )\ ~ p-co& d (? COt ° W + ^ C ° Se ° 2 °) + 4ma ^ = " 4w "^' " ' (5) 219. The case s = 1 includes, as forced oscillations, Laplace's ' Oscillations of the Second Species,' where the disturbing potential is a tesseral harmonic of the second order ; viz. f=iT"sin0cos<9.cosO£ + (£ + e), (1) where cr differs not very greatly from co. This includes the lunar and solar diurnal tides. In the case of a disturbing body whose proper motion could be neglected, we should have a = &>, exactly, and therefore / = \ . In the case of the moon, the orbital motion is so rapid that the actual period of the principal lunar diurnal tide is very appreciably longer than a sidereal day*; but the sup- position that/=|- simplifies the formulae so materially that we adopt it in the following investigation!. We find that it enables us to calculate the forced oscillations when the depth follows the law h = (l-qcos 2 d)h , (2) where q is any given constant. * It is to be remarked, however, that there is an important term in the harmonic development of (2 for which <r — w exactly, provided we neglect the changes in the plane of the disturbing body's orbit. This period is the same for the sun as for the moon, and the two partial tides thus produced combine into what is called the ' luni-solar ' diurnal tide. t Taken with very slight alteration from Airy, "Tides and Waves," Arts. 95 ..., and Darwin, Encyc. Brit. (9th ed.), xxiii. 359. 342 Tidal Waves [chap, viii Taking an exponential factor g l >*+*+ e ) ) and therefore putting s=l,/= J, in Art. 218 (3), and assuming £' = Csin(9cos<9, (3) G we find u = — i<r — , v = a — .cos0 (4) m m Substituting in the equation of continuity (Art. 318 (2)), we get f+?-£g. ■■■; < 5 > which is consistent with the law of depth (2), provided fl— . * , H" (6) 1 — 2qh /ma 1 his gives c = — ■ f ' ; — £ (7) One remarkable consequence of this formula is that in the case of uniform depth (q = 0) there is no diurnal tide, so far as the rise and fall of the surface is concerned. This result was first established (in a different manner) by Laplace, who attached great importance to it as shewing that his kinetic theory was able to account for the relatively small values of the diurnal tide as then (imperfectly) known, in striking contrast to what would be demanded by the equilibrium theory. But, although with a uniform depth there is no rise and fall, there are tidal currents. It appears from (4) that every particle describes an ellipse whose major axis is in the direction of the meridian, and of the same length in all latitudes. The ratio of the minor to the major axis is cos 0, and so varies from 1 at the poles to at the equator, where the motion is wholly N. and S. 220. In the case 5=2, the forced oscillations of most importance are where the disturbing potential is a sectorial harmonic of the second order. These constitute Laplace's 'Oscillations of the Third Species/ for which f = # ,/, sin 2 0.cos(<7*+2(£ + e), (1) where a is nearly equal to 2co. This includes the most important of all the tidal oscillations, viz. the lunar and solar semi-diurnal tides. If the orbital motion of the disturbing body were infinitely slow we should have cr— 2o>, and therefore /= 1; for simplicity we follow Laplace in making this approximation, although it is a somewhat rough one in the case of the principal lunar tide*. A solution similar to that of the preceding Art can be obtained for the special law of depth f h = h o sin 2 (2) * There is, however, a 'luni-solar' semi-diurnal tide whose speed is exactly 2w if we neglect the changes in the planes of the orbits. Cf. p. 341, first footnote, t Cf. Airy and Darwin, 11. cc. 219-221] Semi-diurnal Tide 343 Adopting an exponential factor e ifiu>t+24t+e) , and putting therefore / = 1, 5 = 2, we find that if we assume ?' = Csin 2 0, (3) the equations (3) of Art. 218 give * ff n l a a n 1 + cos2 # / a \ u= — Ccot0, v = -=-C r-3— , (4) m 2m sin v whence, substituting in Art. 218 (2), f=— °.Csin 2 (5) ma x ' Putting f= f' + f, and substituting from (1) and (3), we find C=- _ J, R"\ (6) 1 — 2h /ma and therefore 2Vma g (?) 1 — zho/ma For such depths as actually occur in the ocean 2h <ma, and the tide is therefore inverted. It may be noticed that the formulae (4) make the velocity infinite at the poles, as was to be expected, since the depth there is zero. 221. For any other law of depth a solution can only be obtained in the form of a series. In the case of uniform depth, we find, putting s = 2, / = 1, 4,malh = in Art. 218(5), (1 -^ 2)2 ^ +{/3(1 -^ 2)2 - v - 6 J ?/= -' 8(1 - /a2)2 ^ (8) where //, is written for cos 6. In this form the equation is somewhat intract- able, since it contains terms of four different dimensions in /jl. It simplifies a little, however, if we transform to v, =(1-^)*, = sin <9, as independent variable; viz. we find 1,2(1 ~ v2)d ^' v %~^~ 2p2 ~ ^ 4) f ' = ~ ^ = " ^ E " " 6 ' "* (9) which is of three different dimensions in v. To obtain a solution for the case of an ocean covering the globe, we assume £' = B + B 2 v*+BijA+... + BqvV+ (10) Substituting in (9), and equating coefficients, we find £ = 0, £ 2 = 0, 0.^4 = 0, (11) 1 6B 6 - 10£ 4 + /3#"'=0, (12) and thenceforward 2j(2j + 6)B 2j+i -2j(2j + Z)B 2H2 + {3B 2j = (13) These equations give B Q> B 8} ... B 2 j, ... in succession, in terms of i? 4 , which is so far undetermined. It is obvious, however, from the nature of the 344 Tidal Waves [chap, viii problem, that, except for certain special values of h (and therefore of ft), which are such that there is a free oscillation of corresponding type (s = 2) having the speed 2w, the solution must be unique. We shall see, in fact, that unless B& have a certain definite value the solution above indicated will make the meridian component (u) of the velocity discontinuous at the equator*. The argument is in some respects similar to that of Art. 217. If we denote by Nj the ratio B^^By of consecutive coefficients, we have, from (13), *** 2j+6 2j(2j + 6)iV ( ^> from which it appears that, with increasing j t Nj must tend to one or other of the limits and 1. More precisely, unless the limit of Nj be zero, the limiting form of N j+1 will be (2j+3)/(2; + 6),orl-|, approximately. The latter is identical with the limiting form of the ratio of the coefficients of v 2j and v 2 J~ 2 in the expansion of (1 — v 2 )^. We infer that, unless Bi have such a value as to make N m — 0, the terms of the series (10) will become ultimately comparable with those of (1 — v 2 )%, so that we may write £' = £ + (]. -i^Jf, (15) where L, M are functions of v which do not vanish for v = 1. Near the equator (v = 1) this makes £-*(!-/>»£-** < i6 > Hence, by Art. 218 (3), u would change from a certain definite value to an equal but opposite value as we cross the equator. It is therefore essential, for our present purpose, to choose the value of i? 4 so that N^ = 0. This is effected by the same method as in Art. 217. Writing (13) in the form "•-m^r <"> we see that Nj must be given by the converging continued fraction __£ g B Ar 2j (2; + 6) (2j + 2) (2; + 8) (gt + 4) (2f + 10) '" 2J + 3 2JTTT 2j+7 (18 > 2j + 6 2j + 8 2J+10 * In the case of a polar sea bounded by a small circle of latitude whose angular radius is <\tt, the value of B A is determined by the condition that w = 0, or d^/dv = 0, at the boundary. 22i] Semi-diurnal Tide 345 This holds from j = 2 upwards, but it appears from (12) that it will give also the value of Ni (not hitherto defined), provided we use this symbol for B\\H'" . We have then Finally, writing £=f + f, we obtain qH'" = v 2 + N x v* + N^v* + N x N 2 N*v* + (19) As in Art. 217, the practical method of conducting the calculation is to assume an approximate value for iVj+i, where j is a moderately large number, and then to deduce Nj, iV}_i, ... N 2i A 7 i in succession by means of the formula (17). The above investigation is taken substantially from the very remarkable paper written by Kelvin* in vindication of Laplace's treatment of the problem, as given in the Mecanique Celeste. In the passage more especially in question, Laplace determines the constant J5 4 by means of the continued fraction for iVj, without, it must be allowed, giving any adequate justification of the step ; and the soundness of this procedure had been disputed by Airyt, and after him by Ferrel J. Laplace, unfortunately, was not in the habit of giving specific references, so that few of his readers appear to have become acquainted with the original presentment § of the kinetic theory, where the solution for the case in question is put in a very convincing, though somewhat different, form. Aiming in the first instance at an approximate solution by means of & finite series, thus : C = B iV * + B 6 ve+... + B 2k+2 v™ + *, (20) Laplace remarks j| that in order to satisfy the differential equations, the coefficients would have to fulfil the conditions \6B 6 -10B i +l3H'" = 0, \ 40£ 8 -285 6 + /3£ 4 =0, ,(21) (2* -2) (2£ + 4) B 2k + 2 -(2k-2) (2k + 1) B 2k + pB 21c _ 2 = 0,\ - 2k (2£ + 3) B 2k + 2 +(3B 2k =0, j (3B 2k + 2 = 0,l as is seen at once by putting B 2k + 4 = 0, B 2k + Q = 0, ... in the general relation (13). We have here k + l equations between k constants. The method followed is to determine the constants by means of the first k relations ; we thus obtain an exact solution, not of the proposed differential equation (9), but of the equation as modified by the addition of a term @B 2k + 2 v' 2k + 6 to the right-hand side. This is equivalent to an alteration of the disturbing force, and if we can obtain a solution such that the required alteration is very small, we may accept it as an approximate solution of the problem in its original form IT. * Sir W. Thomson, "On an Alleged Error in Laplace's Theory of the Tides," Phil. Mag. (4), 1. 227 (1875) [Papers, iv. 231]. f "Tides and Waves," Art. 111. X "Tidal Eesearches," U.S. Coast Survey Rep. 1874, p. 154. § "Becherches sur quelques points du systeme du monde," Mem. de V Acad. roy. des Sciences, 1776 [1779] [Oeuvres, ix. 187...]. || Oeuvres, ix. 218. The notation has been altered. II It is remarkable that this argument is of a kind constantly employed by Airy himself in his researches on waves. 346 Tidal Waves [chap. VIII Now, taking the first k relations of the system (21) in reverse order, we obtain B 2k + 2 in terms of 2? 2fc , thence B 2 k in terms of i>2&-i> an ^ so on > until, finally, 2? 4 is expressed in terms of H"' ; and it is obvious that if k be large enough the value of 2? 2 fc + 2> ana tne consequent adjustment of the disturbing force which is required to make the solution exact, will be very small. This will be illustrated presently, after Laplace, by a numerical example. The process just given is plainly equivalent to the use of the continued fraction (18) in the manner already explained, starting with j+l = k, and iy£=/3/2&(2&+3). The continued fraction, as such, does not, however, make its appearance in the memoir here referred to, but was introduced in the Mecanique Celeste, probably as an after-thought, as a condensed expression of the method of computation originally employed. The table below gives the numerical values of the coefficients of the several powers of v in the formula (19) for ?/#'", in the cases /3 = 40, 20, 10, 5, 1, which correspond to depths of 7260, 14520, 29040, 58080, 290400 feet, respectively*. The last line gives the value of £/H"' for i> = l, i.e. the ratio of the amplitude at the equator to its equilibrium-value. At the poles (v = 0), the tide has in all cases the equilibrium- value zero. /3 = 40 = 20 = 10 = 5 = 1 v i + l'OOOO f- 1 -oooo ■t- 1 -oooo + 1-0000 + 1-0000 V* + 20-1862 -0-2491 +6-1915 + 0-7504 +0-1062 V* + 10-1164 -1-4056 + 3-2447 + 0-1566 + 0-0039 V 8 -13-1047 -0-8594 + 0-7234 + 0-0157 + 0-0001 „io - 15-4488 -0-2541 + 0-0919 + 0-0009 „12 - 7-4581 - 0-0462 + 0-0076 V™ - 2-1975 -0-0058 + 0-0004 „16 - 0-4501 - 0-0006 V 8 - 0-0687 „20 - 0-0082 „ 22 - 0-0008 „24 - 0-0001 - 7-434 -1-821 + 11-259 + 1-924 + 1-110 We may use the above numerical results to estimate the closeness of the approxi- mation in each case. For example, when /3 = 40, Laplace finds B^= — •000004Z/'"' ; the addition to the disturbing force which is necessary to make the solution exact would then be - •00002/T"i/ 30 , and would therefore bear to the actual force the ratio - '0C002I/ 28 . It appears from (19) that near the poles, where v is small, the tides are in all cases direct. For sufficiently great depths, /5 will be very small, and the formulae (17) and (19) then shew that the tide has everywhere sensibly the equilibrium-value, all the coefficients being small except the first, which is unity. As h is diminished, /3 increases, and the formula (17) shews that each of the ratios iV} will continually increase, except when it changes sign * The first three cases were calculated by Laplace, I.e. ante p. 330 ; the last by Kelvin. The numbers relating to the third case have been slightly corrected, in accordance with the computa- tions of Hough ; see p. 347. 221-222] Hough's Theory 347 from + to — by passing through the value oo . No singularity in the solution attends this passage of A 7 } through oo , except in the case of N% t since, as is easily seen, the product Nj^Nj remains finite, and the coefficients in (19) are therefore all finite. But when jVi=oo, the expression for f becomes infinite, shewing that the depth has then one of the critical values already referred to. The table on p. 346 indicates that for depths of 29040 feet, and upwards, the tides are everywhere direct, but that there is some critical depth between 29040 feet and 14520 feet, for which the tide at the equator changes from direct to inverted. The largeness of the second coefficient in the case /3 = 40 indicates that the depth could not be reduced much below 7260 feet before reaching a second critical value. Whenever the equatorial tide is inverted, there must be one or more pairs of nodal circles (f=0), symmetrically situated on opposite sides of the equator. In the case of /3 = 40, the position of the nodal circles is given by v = '95, or 6 = 90° ± 18°, approximately*. 222. The dynamical theory of the tides, in the case of an ocean covering the globe, with depth uniform along each parallel of latitude, has been greatly improved and developed by Hough f, who, taking up an abandoned attempt of Laplace, substituted expansions in spherical harmonics for the series of powers of /z, (or v). This has the advantage of more rapid convergence, especially, as might be expected, in cases where the influence of the rotation is relatively small; and it also enables us to take account of the mutual attraction of the particles of water, which, as we have seen in the simpler problem of Art. 200, is by no means insignificant. If the surface-elevation f, and the conventional equilibrium tide-height f (in which the effect of mutual attraction is not included), be expanded in series of spherical harmonics, thus t-sc, g-s?. (i) the complete expression for the disturbing potential will be cf. Art. 200. The series on the right hand is to be substituted for f in the equations of Arts. 214...; this will be allowed for if we write r = SKf,-f n ), (2) o where a n = l - -£ , (3) zn + 1 po in modification of the notation of Art. 215 (5) or Art. 218 (4). * For a fuller discussion of these points reference may be made to the original investigation of Laplace, and to Kelvin's papers. f "On the Application of Harmonic Analysis to the Dynamical Theory of the Tides," Phil. Trans. A, clxxxix. 201, and cxci. 139 (1897). See also Darwin's Papers, i. 349. 348 Tidal Waves [chap, viii In the oscillations of the 'First Species,' the differential equation may be written «(£$©+«-• <•» If we assume S=XC n P n (fi,), t-27„P n (/*), (5) we have ?' = 2 (a» C» - 7n) Pn 00 (6) Substituting in (4), and integrating between the limits — 1 and /jl, we find 2 (a. G n - 7n ) (1 - ft *jg + 2/3C„ {(/ 2 - 1) + (1 - ?*)} £*.*• = 0. . . .(7) Now, by known formulae of zonal harmonics*, /> 1 and J P n cfy* = 2y| +1 (Pn+i - iVi) ! /^Pn+2 _ CLP^\ _ 1 (dPn _ dP n-2 \ U ) 2/i + l [2n + 3 \ c^yLt rfyit / 2n — 1 \ a(/-t rf/* _1 ^Pn+2 2 rfP„ (2w + l)(2»+3) rf/i (2n-l)(2» + 3) d/* j 1 ^Pn-2 /qx (2w-l)(2» + l) (fy* ' '" W Substituting in (7), and equating to zero the coefficient of (1 — fx 2 ) -=-^ we find Cn+t-LnCn+r - g^g ~C n _ 2 =^, ...(10) (2rc + 3)(2n + 5) n+ ' n n (27i-3)(2n-l)^- 2 ~y3 ' Where X - = ^^) + (2n-l) 2 (2, + 3) -? (U) The relation (10) will hold from n = 1 onwards, provided we put C-i=3, C = 0. The further theory is based substantially on the argument of Laplace, given in Art. 221; and the work follows much the same lines as in Arts. 216, 217, 221. In the free oscillations we have y n = 0, and the admissible values of f are determined by the transcendental equation _1 1 5.7*.99.11M3 X2 ~"Lr^"x 6 -&c. -°' (12) 1 1 T 3.5 2 .77.9 2 .11 A /1QX ^-^^L^r^ (13) * See Todhunter, Functions of Laplace, &c. c. v. ; Whittaker and Watson, Modern Analysis, p. 306. 222-223] Case of Symmetry ; Free Periods 349 according as the mode is symmetrical or asymmetrical with respect to the equator. Alternative forms of the period equations are given by Hough, suitable for computation of the higher roots, and it is shewn that close approximations are given by the equations L n = or 3 p\ gh 2 = l+n(n + l) 1 (2ft-l)(2n + 3) ...(14) 4o) 2 * ' '" v " ' *"' IV* 2n+ 1 p /4<a> 2 a 2 except for the first two or three values of n * The following table gives the periods (in sidereal time) of the slowest symmetrical oscillation (i.e. the one in which the surface-elevation would vary as Pi(ii) if there were no rotation), corresponding to various depths f. F Depth 0* Period Period p (feet) 4w2 h. m. when u = h. m. 40 7260 •44155 18 3-5 32 49 20 14520 •62473 15 11-0 23 12 io ! 29040 •92506 12 28-6 16 25 5 1 1 58080 1-4785 9 52-1 11 35 The results obtained for the forced oscillations of the ' First Species 5 are very similar to those of Art. 217. The limiting form of the long-period tides when o-=0 shews the following results : i 1 p 1 p/p =-181 p/p = Pole Equator Pole Equator 40 •140 •426 •154 •455 20 •266 •551 10 •443 •681 •470 •708 5 •628 •796 •651 •817 The second and third columns give the ratio of the polar and equatorial tides to the respective equilibrium- values]:. The numbers in the fourth and fifth columns are repeated from Art. 217. The comparison shews the effect of the mutual gravitation of the water in reducing the amplitude. 223. In the more general case, where symmetry about the axis is not imposed, the surface-elevation f is expanded by Hough in a series of tesseral harmonics of the type P n s (rie^ t+8 + + * (1) * Eeference may also be made to Poole, Proc. Lond. Math. Soc. (2) xix. 299. + The slowest asymmetrical mode has a much longer period. It involves a displacement of the centre of mass of the water, so that a correction would be necessary if the nucleus were free ; cf. Art. 199. X The numbers are deduced from Hough's results. The paper referred to contains discussions of other interesting points, including an examination of cases of varying depth, with numerical illustrations. 350 Tidal Waves [chap, vm In relation to tidal theory the most important cases are where the disturbing potential is of the form (1), with n = 2 and 5=1 or s = 2. The calculations are necessarily somewhat intricate*, and it may suffice here to mention a few of the more interesting results, which will indicate how the gaps in the previous investigations have been filled. To understand the nature of the free oscillations, it is best to begin with the case of no rotation (« = 0). As co is increased, the pairs of numerically equal, but oppositely signed, values of a which were obtained in Art. 199 begin to diverge in absolute value, that being the greater which has the same sign with g>. The character of the fundamental modes is also gradually altered. These oscillations are distinguished as ' of the First Class.' At the same time certain steady motions which are possible, without change of level, when there is no rotation, are converted into long-period oscillations with change of level, the speeds being initially comparable with &>. The corresponding modes are designated as 'of the Second Class 'f ; cf. Art. 206. The following table gives the speeds of those modes of the First Class which are of most importance in relation to the diurnal and semi-diurnal tides, respectively, and the corresponding periods, in sidereal time. The last column repeats the corresponding periods in the case of no rotation, as calculated from the formula (15) of Art. 200. Second Species [8 = 1] Third Species = 2] Depth (feet) 0} Period h. m. (a Period h. m. Period when w = h. m. 7260 14520 29040 58080 1-6337 - 0-9834 1-8677 -1-2450 2-1641 -1-6170 2-6288 -2-1611 14 41 24 24 12 51 19 16 11 5 14 50 9 8 11 6 1-3347 -0-6221 1-6133 -0-8922 1-9968 -1-2855 2-5535 -1-8575 17 59 38 34 14 52 26 54 12 1 18 40 9 24 12 55 [ 32 49 I 23 12 I 16 25 I 11 35 The quickest oscillation of the Second Class has in each case a period of over a day ; and the periods of the remainder are very much longer. * A simplification is made by Love, "Notes on the Dynamical Theory of the Tides," Proc. Lond. Math. Soc. (2) xii. 309 (1913). He writes dx d\p dx 5^ add a sin >d<f> add cf. Art. 154 (1). The values of x> ^ are expanded in series of spherical harmonics. | These two classes of oscillations have been already encountered in the plane problem of Art. 212. 223] Diurnal and Semi-diurnal Tides 351 As regards the forced oscillations of the ' Second Species,' Laplace's conclusion that when a — a, exactly, the diurnal tide vanishes in the case of uniform depth, still holds. The computation for the most important lunar diurnal tide, for which ajw — '92700, shews that with such depths as we have considered the tides are small compared with the equilibrium heights, and are in the main inverted. Of the forced oscillations of the 'Third Species,' we may note first the case of the solar semi-diurnal tide, for which cr= 2co with sufficient accuracy. For the four depths given in our tables, the ratio of the dynamical tide-height to the conventional equilibrium tide-height at the equator is found to be + 7-9548, -1-5016, -234-87, +2*1389, respectively. " The very large coefficients which appear when hg/4xo 2 a 2 = -^ indicate that for this depth there is a period of free oscillation of semi-diurnal type whose period differs but slightly from half-a-day. On reference to the tables ... it will be seen that we have, in fact, evaluated this period as 12 hours 1 minute, while for the case %/4o> 2 a 2 = -^ we have found a period of 12 hours 5 minutes*. We see then that though, when the period of forced oscillation differs from that of one of the types of free oscillation by as little as one minute, the forced tide may be nearly 250 times as great as the corresponding equilibrium tide, a difference of 5 minutes between these periods will be sufficient to reduce the tide to less than ten times the corresponding equilibrium tide. It seems then that the tides will not tend to become excessively large unless there is very close agreement with the period of one of the free oscillations. " The critical depths for which the forced tides here treated of become infinite are those for which a period of free oscillation coincides exactly with 12 hours. They may be ascertained by putting [a = 2w] in the period- equation for the free oscillations and treating this equation as an equation for the determination of h The two largest roots are..., and the corre- sponding critical depths are about 28,182 feet and 7375 feet "It will be seen that in three cases out of the four here considered the effect of the mutual gravitation of the waters is to increase the ratio of the tide to the equilibrium tide [cf. Art. 221]. In two of the cases the sign is also re- versed. This of course results from the fact that whereas when [p/pi = 0*18093] one of the periods of free oscillation is rather greater than 12 hours, when [p/pi = 0] the corresponding period will be less than 12 hours f." Hough has also computed the lunar semi-diurnal tides for which £-= 0*96350. 2(0 * [Belonging to a mode which comes next in sequence to the one having a period of 17 h. 59 m.] f Hough, Phil. Trans. A, cxci. 178, 179. 352 Tidal Waves [chap, viii For the four depths aforesaid the ratios of the equatorial tide-heights to their equilibrium-values are found to be -2-4187, -1-8000, +110725, +1*9225, respectively. "On comparison of these numbers with those obtained for the solar tides..., we see that for a depth of 7260 feet the solar tides will be direct while the lunar tides will be inverted, the opposite being the case when the depth is 29,040 feet. This is of course due to the fact that in each of these cases there is a period of free oscillation intermediate between twelve solar (or, more strictly, sidereal) hours and twelve lunar hours. The critical depths for which the lunar tides become infinite are found to be 26,044 feet and 6448 feet. "Consequently this phenomenon will occur if the depth of the ocean be between 29,182 feet and 26,044 feet, or between 7375 feet and 6448 feet. An important consequence would be that for depths lying between these limits the usual phenomena of spring and neap tides would be reversed, the higher tides occurring when the moon is in quadrature, and the lower at new and full moon* " 223 a. Some important contributions to the dynamical theory have been made by Goldsbrough. Considering, first, the tides in an ocean of uniform depth bounded by one or two parallels of latitude, he finds, in the case of a polar basin of angular radius 30°, for instance, that for such depths as have been considered in Arts. 217, 221 the long-period tides and the semi-diurnal tides do not deviate very widely from the values given by the equilibrium theory, when this is corrected as explained in the Appendix f. The case is different with the diurnal tides, which vary considerably with the size of the basin and the depth, and are as a rule considerable, whereas we have seen that in a uniform ocean covering the globe they are negligible. In the case of an equatorial belt{, the long-period tides again approximate to the equilibrium values, whilst the diurnal and semi-diurnal deviate widely, to an extent which varies considerably with the latitudes of the boundaries. The variations here met with are doubtless conditioned by the relation between the imposed period and the natural periods of free oscillation. This question has been examined by Goldsbrough with reference to the semi-diurnal tides of the Atlantic ocean, which forms a more or less limited and isolated system. Taking the case of an ocean limited by two meridians 60° apart, and assuming the law of depth h = h sin 2 6, * Hough, I.e., where reference is made to Kelvin's Popular Lectures and Addresses, London, 1894, ii. 22 (1868). t Proc. Lond. Math. Soc. (2) xiv. 31 (1913). J Ibid. xiv. 207 (1914). 223-224] Semi-diurnal Tide 353 he finds* that there will be a free oscillation with <r = 2&> exactly, provided ho = 23,200 ft., which means a mean depth of 15,500 ft. With ho = 25,320 ft., or a mean depth of 16,880 ft., he finds that the forced tides of the above period are still very large compared with the equilibrium values. In a more recent paper f by Goldsbrough and Colborne the depth is taken to be uniform and equal to the estimated mean depth (12,700 ft.) of the Atlantic. For the imposed frequency they take that of the principal semi- diurnal constituent (usually denoted by M 2 ) of the lunar disturbing force (cr/2o> = *9625). The amplitudes, though not so great as before, prove to be largely in excess of the equilibrium values. The diurnal tide in an ocean of this type has been investigated by Colborne j. 224. It is not easy to estimate, in any but the most general way, the extent to which the foregoing conclusions of the dynamical theory would have to be modified if account could be taken of the actual configuration of the ocean, with its irregular boundaries and irregular variation of depth§. One or two points may however be noticed. In the first place, the formulae (1) of Art. 206 would lead us to expect for any given tide a phase-difference, variable from place to place, between the tide-height and the disturbing force ||. Thus, in the case of the lunai semi-diurnal tides, for example, high-water or low-water need not synchronize with the transit of the moon or anti-moon across the meridian. More precisely, in the case of a disturbing force of given type for which the equilibrium tide-height at a particular place would be f = a cos at, (1) the dynamical tide-height will be £=A cos (at -e), (2) where the ratio A /a, and the phase-difference e, will be functions of the speed a, as well as of the position of the station. Again, consider the superposition of two oscillations of the same type but of slightly different speeds, e.g. the lunar and solar semi-diurnal tides. If the origin of t be taken at a syzygy, we have f = a cos at + a' cos at, (3) and £=A cos(at-e) + A' cos (a't-e) (4) This may be written f = ( A + A' cos <£) cos (at — e) + A' sin </> sin (at - e), (5) where = (a - a) t- e + e' (6) * Proc. Roy. Soc. A, cxvii. 692 (1927). t Ibid, cxxvi. 1 (1929). + Ibid, cxxxi. 38 (1931). § As to the general mathematical problem reference may be made to Poincare, " Sur l'equi- libre et les mouvements des mers," Liouville (5), ii. 57, 217 (1896), and to his Legons de mecanique celeste, iii. || This is illustrated by the canal problem of Art. 184. 354 Tidal Waves [chap, viii If the first term in the second member of (4) represents the lunar, and the second the solar tide, we shall have a < a, and A > A'. If we write A + A' cos <f> = C cos a, A' sin </> = (7 sin a, (7) we get f=Ccos(V£-e~a), (8) where C = (A* + 2AA' cos d> + A' 2 )*, a = tan" 1 ^' sin * . . . .( 9) v T A + J. cos <f> This may be described as a simple-harmonic oscillation of slowly varying amplitude and phase. The amplitude ranges between the limits A ± A', whilst a may be supposed to lie always between ±%ir. The 'speed' must also be regarded as variable, viz. we fiad d, 4 x <tA 2 + (*+</) A A' cos $ + *' A' 2 „ m S^- 41 )- A*+ZAA> G o^Ia>* ( 10 > This ranges between Aa + A'a , Act — A' a' /,iv* A +4' and Z^T (U) The above is the well-known explanation of the phenomena of the spring- and neap- tides f; but we are now concerned further with the question of phase. On the equilibrium theory, the maxima of the amplitude G would occur whenever {a —a)t = 2n7r, where n is integral. On the dynamical theory the corresponding times of maximum are given by O' - a) *-(€'-€) = 2nir, i.e. the dynamical maxima follow the statical by an interval J (*' -*)/(</-<,). If the difference between a and <r were infinitesimal, this would be equal to de/da. The fact that the time of high-water, even at syzygy, may follow or precede the transit of the moon or anti-moon by an interval of several hours is well known §. The interval, when reckoned as a retardation, is, moreover, usually greater for the solar than for the lunar semi-diurnal tide, with the result that the spring-tides are in many places highest a day or two after the corresponding syzygy. The latter circumstance has been ascribed || to the operation of Tidal Friction (for which see Chapter xi.), but it is evident * Helmholtz, Lehre von den Tonempjindungen (2 9 Aufl.), Braunschweig, 1870, p. 622. t Cf . Thomson and Tait, Art. 60. X This interval may of course be negative. § The values of the retardations (which we have denoted by e) for the various tidal com- ponents, at a number of ports, are given by Baird and Darwin, "Kesults of the Harmonic Analysis of Tidal Observations," Proc. R. S. xxxix. 135 (1885), and Darwin, " Second Series of Results...," Proc. R. S. xlv. 556 (1889). || Airy, "Tides and Waves," Art. 459. 224-225] Lag of the Tides 355 that the phase-differences which are incidental to a complete dynamical theory, even in the absence of friction, cannot be ignored in this connection. There is reason to believe that they are, indeed, far more important than those due to the latter cause. Lastly, it was shewn in Arts. 206, 217 that the long-period tides may deviate very considerably from the values given by the equilibrium theory, owing to the possibility of certain steady motions in the absence of disturbance. It has been pointed out by Rayleigh* that these steady motions may be impossible in certain cases where the ocean is limited by perpendicular barriers. Referring to Art. 214 (6), it appears that if the depth h be uniform, f must (in the steady motion) be a function of the co-latitude 6 only, and therefore by (4) of the same Art., the eastward velocity v must be uniform along each parallel of latitude. This is inconsistent with the existence of a perpendicular barrier extending along a meridian. The objection would not necessarily apply to the case of a sea shelving gradually from the central parts to the edgef. 225. We may complete the investigation of Art. 200 by a brief notice of the question of the stability of the ocean, in the case of rotation. It has been shewn in Art. 205 that the condition of secular stability is that V — T should be a minimum in the equilibrium configuration. If we neglect the mutual attraction of the elevated water, the application to the present problem is very simple. The excess of the quantity V — T over its undisturbed value is evidently fj{jV-i«'-*) «**}<*$ W where M* denotes the potential of the earth's attraction, 8S is an element of the oceanic surface, and the rest of the notation is as before. Since M* - ^co 2 ^ 2 is constant over the undisturbed level (z = 0), its value at a small altitude z may be taken to be gz + const., where, as in Art. 213, 9 = d (v-4«V)l (2) dz iz =0 Since ff£dS = 0, on account of the constancy of volume, we find from (1) that the increment of V — T Q is iffgPdS (3) This is essentially positive, and the equilibrium is therefore 'secularly' stable J. * "Note on the Theory of the Fortnightly Tide," Phil. Mag. (6) v. 136 (1903) [Papers, iv. 84]. f The theory of the limiting forms of long-period tides in oceans of various types is discussed by Proudman, Proc. Lond. Math. Soc. (2) xiii. 273 (1913). X Cf. Laplace, Mecanique Celeste, Livre 4 me , Arts. 13, 14. 356 Tidal Waves [chap, vm It is to be noticed that this proof does not involve any restriction as to the depth of the fluid, or as to smallness of the ellipticity, or even as to symmetry of the undisturbed surface with respect to the axis of rotation. If we wish to take into account the mutual attraction of the water, the problem can only be solved without difficulty when the undisturbed surface is nearly spherical, and we neglect the variation of g. The question (as to secular stability) is then exactly the same as in the case of no rotation. The calculation for this case will find an appropriate place in the next chapter (Art. 264). The result, as we might anticipate from Art. 200, is that the necessary and sufficient condition of stability of the ocean is that its density should be less than the mean density of the earth*. 226. This is perhaps the most suitable occasion for a few additional remarks on the general question of stability of dynamical systems. We have in the main followed the ordinary usage which pronounces a state of equilibrium, or of steady motion, to be stable or unstable according to the character of the solution of the approximate equations of disturbed motion. If the solution consists of series of terms of the type Ge ±u } where all the values of X are pure imaginary (i.e. of the form ia), the undisturbed state is usually reckoned as stable; whilst if any of the Vs are real, it is accounted unstable. In the case of disturbed equilibrium, this leads algebraically to the usual criterion of a minimum value of V as a necessary and sufficient condition of stability. It has in recent times been questioned whether this conclusion is, from a practical point of view, altogether warranted. It is pointed out that since the approximate dynamical equations become less and less accurate as the deviation from the equilibrium configuration increases, it is a matter for examination how far rigorous conclusions as to the ultimate extent of the deviation can be drawn from themf. The argument of Dirichlet, which establishes that the occurrence of a minimum value of V is a sufficient condition of stability, in any practical sense, has already been referred to. No such simple proof is available to shew without qualification that this condition is necessary. If, however, we recognize the existence of dissipative forces, which are called into play by any motion whatever of the system, the conclusion can be drawn as in Art. 205. A little consideration will shew that a good deal of the obscurity which attaches to the question arises from the want of a sufficiently precise mathematical definition of what is meant by 'stability.' The difficulty is encountered in an aggravated form when we pass to the question of * Cf. Laplace, I.e. t See papers by Liapounoff and Hadamard, Liouville (5), iii. (1897). 225-226] Stability of the Ocean 357 stability of motion. The various definitions which have been propounded by different writers are examined critically by Klein and Sommerfeld in their book on the theory of the top*. Rejecting previous definitions, they base their criterion on the character of the changes produced in the path of the system by small arbitrary disturbing impulses. If the undisturbed path be the limiting form of the disturbed path when the impulses are indefinitely diminished, it is said to be stable, but not otherwise. For instance, the vertical fall of a particle under gravity is reckoned as stable, although for a given impulsive disturbance, however small, the deviation of the particle's position at any time t from the position which it occupied in the original motion increases indefinitely with t. Even this criterion, as the writers referred to themselves recognize, is not free from ambiguity unless the phrase 1 limiting form,' as applied to a path, be strictly defined. It appears moreover that a definition which is analytically precise may not in all cases be easy to reconcile with geometrical prepossessions f. The foregoing considerations have reference, of course, to the question of 'ordinary' stability. The more important theory of 'secular' stability (Art. 205) is not affected. We shall meet with the criterion for this, under a somewhat modified form, at a later stage in our subject J. * Ueber die Theorie des Kreisels, Leipzig, 1897..., p. 342. f Some good illustrations are furnished by Particle Dynamics. Thus a particle moving in a circle about a centre of force varying inversely as the cube of the distance will if slightly disturbed either fall into the centre, or recede to infinity, after describing in either case a spiral with an infinite number of convolutions. Each of these spirals has, analytically, the circle as its 'limiting form,' although the motion in the latter is most naturally described as unstable. Cf. Korteweg, Wiener Ber. May 20, 1886. A narrower definition has been given by Love, and applied by Bromwich to several dynamical and hydrodynamical problems; see Proc. Lond. Math. Soc. (1) xxxiii. 325 (1901). X This summary is taken substantially from the Art. "Dynamics, Analytical," in Encyc. Brit. 10th ed. xxvii. 566 (1902), and 11th ed. viii. 756 (1910). APPENDIX TO CHAPTER VIII ON TIDE-GENERATING FORCES a. If, in the annexed figure, and C be the centres of the earth and of the disturbing body (say the moon), the potential of the moon's attraction at a point P near the earth's surface will be — yM/CP, where M denotes the moon's mass, and y the gravitation- constant. If we put OC=D, OP=r, and denote the moon's (geocentric) zenith-distance at P, viz. the angle POC, by $, this potential is equal to yM (D 2 -2rDcos$ + r 2 )b' P We require, however, not the absolute accelerative effect at P, but the acceleration relative to the earth. Now the moon produces in the whole mass of the earth an acceleration yM/D 2 * parallel to 0(7, and the potential of a uniform field of force of this intensity is evidently _JL_. rC os& Subtracting this from the former result we get, for the potential of the relative attraction at % Q== yM ^.roosS (1) (Z) 2 -*2ri)cos#+r 2 )* ^ 2 This function Q is identical with the ' disturbing-function ' of planetary theory. Expanding in powers of r/D, which is in our case a small quantity, and retaining only the most important term, we find G=i^ 2 (i-cos^) ( 2 ) Considered as a function of the position of P, this is a zonal harmonic of the second degree, with OC as axis. The reader will easily verify that, to the order of approximation adopted, Q. is equal to the joint potential of two masses, each equal to \M, placed, one at (7, and the other at a point C in CO produced such that 0C' = 0Cf. h. In the 'equilibrium-theory' of the tides it is assumed that the free surface takes at each instant the equilibrium-form which might be maintained if the disturbing body were to retain unchanged its actual position relative to the rotating earth. In other * The effect of this is to produce a monthly inequality in the motion of the earth's centre about the sun. The amplitude of the inequality in radius vector is about 3000 miles; that of the inequality in longitude is about 7" ; see Laplace, Mecanique Celeste, Livre 6 me , Art. 30, and Livre 13 me , Art. 10. f Thomson and Tait, Art. 804. These two fictitious bodies are designated as 'moon' and 'anti-moon,' respectively. Equilibrium Theory 359 words, the free surface is assumed to be a level-surface under the combined action of gravity, of centrifugal force, and of the disturbing force. The equation to this level- surface is *-|o) 2 o; 2 + Q = const., (3) where o> is the angular velocity of the rotation, w denotes the distance of any point from the earth's axis, and M> is the potential of the earth's attraction. If we use square brackets [ ] to distinguish the values of the enclosed quantities at the undisturbed level, and denote by £ the elevation of the water above this level due to the disturbing potential Q, the above equation is equivalent to [¥ - £a> 2 G7 2 ] + f"g- (¥ - |a> 2 OT 2 )l C+ Q =COnst., •(4) approximately, where dfdz is used to indicate a space-differentiation along the normal outwards. The first term is of course constant, and we therefore have f--J+4 <*) .(6) where, as in Art. 213, g— ^- (¥— ^co 2 ar 2 ) Evidently, g denotes the value of l apparent gravity ' ; it will of course vary more or less with the position of P on the earth's surface. It is usual, however, in the theory of the tides, to ignore the slight variations in the value of g, and the effect of the ellipticity of the undisturbed level on the surface- value of Q. Putting, then, r=a, g = yE/a 2 , where E denotes the earth's mass, and a the mean radius of the surface, we have, from (2) and (5), C=H (cos* S-D + C, (7) where H== ^'~E'\^) ' a > ^ as in Art. 180. Hence the equilibrium-form of the free surface is a harmonic spheroid of the second order, of the zonal type, whose axis passes through the disturbing body. C. Owing to the diurnal rotation, and also to the orbital motion of the disturbing body, the position of the tidal spheroid relative to the earth is continually changing, so that the level of the water at any particular place will continually rise and fall. To analyse the character of these changes, let be the co-latitude, and <f> the longitude, measured eastward from some fixed meridian, of any place P, and let A be the north-polar- distance, and o the hour-angle west of the same meridian, of the disturbing body. We have, then, cos£ = cos A cos + sin Asintf cos (a + (f>), (9) and thence, by (7), C= f H (cos 2 A - 1) (cos 2 6-1) + \H sin 2 A sin 20 cos (a + <£) + \H sin 2 A sin 2 cos 2 (a + <£) + C. (10) Each of these terms may be regarded as representing a partial tide, and the results superposed. Thus, the first term is a zonal harmonic of the second order, and gives a tidal spheroid vmmetrical with respect to the earth's axis, having as nodal lines the parallels for which cos 2 = ^, or = 90° ±35° 16'. The amount of the tidal elevation in any particular latitude varies as cos 2 A — ^. In the case of the moon the chief fluctuation in this quantity has a period of about a fortnight; we have here the origin of the 'lunar fortnightly' or ( declinational ' tide. When the sun is the disturbing body, we have a ' solar semi-annual ' tide. It is to be noticed that the mean value of cos 2 A — J with respect to the time is not 360 Appendix to Chapter VIII zero, so that the inclination of the orbit of the disturbing body to the equator involves as a consequence a permanent change of mean level. Cf. Art. 183. The second term in (10) is a spherical harmonic of the type obtained by putting n = 2, s = l in Art. 86 (7). The corresponding tidal spheroid has as nodal lines the meridian which is distant 90° from that of the disturbing body, and the equator. The disturbance of level is greatest in the meridian of the disturbing body, at distances of 45° N. and S. of the equator. The oscillation at any one place goes through its period with the hour- angle, a, i.e. in a lunar or solar day. The amplitude is, however, not constant, but varies slowly with A, changing sign when the disturbing body crosses the equator. This term accounts for the lunar and solar ' diurnal ' tides. The third term is a sectorial harmonic (n = 2, -5=2), and gives a tidal spheroid having as nodal lines the meridians which are distant 45° E. and W. from that of the disturbing body. The oscillation at any one place goes through its period with 2a, i.e. in half a (lunar or solar) day, and the amplitude varies as sin 2 A, being greatest when the disturbing body is on the equator. We have here the origin of the lunar and solar ' semi-diurnal ' tides. The 'constant' C is to be determined by the consideration that, on account of the invariability of volume, we must have SJ£dS=0, (11) where the integration extends over the surface of the ocean. If the ocean cover the whole earth we have (7=0, by the general property of spherical surface -harmonics quoted in Art. 87. It appears from (7) that the greatest elevation above the undisturbed level is then at the points 5=0, 5 = 180°, i.e. at the points where the disturbing body is in the zenith or nadir, and the amount of this elevation is §//". The greatest depression is at places where 5 = 90°, i.e. the disturbing body is on the horizon, and is ^ff. The greatest possible range is therefore equal to IT. In the case of a limited ocean, C does not vanish, but has at each instant a definite value depending on the position of the disturbing body relative to the earth. This value may be easily written down from equations (10) and (11); it is a sum of spherical harmonic functions of A, a, of the second order, with constant coefficients in the form of surface-integrals whose values depend on the distribution of land and water over the globe. The changes in the value of C, due to relative motion of the disturbing body, give a general rise and fall of the free surface, with (in the case of the moon) fortnightly, diurnal, and semi-diurnal periods. This ' correction to the equilibrium-theory ' as usually presented, was first fully investigated by Thomson and Tait* The necessity for a correction of the kind, in the case of a limited sea, had however been recognized by D. Bernoulli t. The correction has an influence on the time of high water, which is no longer synchronous with the maximum of the disturbing potential. The interval, moreover, by which high water is accelerated or retarded differs from place to place J. d. We have up to this point neglected the mutual attraction of the particles of the water. To take this into account, we must add to the disturbing potential Q, the gravitation-potential of the elevated water. In the case of an ocean covering the earth, the correction can be easily applied, as in Art. 200. If we put n = 2 in the formulae of * Natural Philosophy, Art. 808; see also Darwin, "On the Correction to the Equilibrium Theory of the Tides for the Continents," Proc. Roy. Soc. April 1, 1886 [Papers, i. 328]. It appears as the result of a numerical calculation by Prof. H. H. Turner, appended to this paper, that with the actual distribution of land and water the correction is of little importance. t Traite sur le Flux et Reflux de la Mer, c. xi. (1740). This essay, as well as the one by Maclaurin cited on p. 307, and another on the same subject by Euler, is reprinted in Le Seur and Jacquier's edition of Newton's Principia. % Thomson and Tait, Art. 810. The point is illustrated by the formula (3) of Art. 184 supra. Harmonic Analysis 361 that Art., the addition to the value of Q is — § p/p . gC ; and we thence find without difficulty ?=i4k<»**-» (12) It appears that all the tides are increased, in the ratio (1 - fp/p ) _1 - If we assume p/p = -18, this ratio is 1*12. e. So much for the equilibrium theory. For the purposes of the kinetic theory of Arts. 213-224, it is necessary to suppose the value (10) of £ to be expanded in a series of simple-harmonic functions of the time. The actual expansion, taking account of the variations of A and a, and of the distance D of the disturbing body (which enters into the value of H), is a somewhat complicated problem of Physical Astronomy, into which we do not enter* Disregarding the constant C, which disappears in the dynamical equations (1) of Art. 215, the constancy of volume being now secured by the equation of continuity (2), it is easily seen that the terms in question will be of three distinct types. First, we have the tides of long period, for which ~£= H' (cos 2 d-%) . cos (o-t + e). .... (13) The most important tides of this class are the ' lunar fortnightly ' for which, in degrees per mean solar hour, o- = l o, 098, and the 'solar-annual' for which o- = 0°*082. Secondly, we have the diurnal tides, for which (=H"sm0cosd .cos(o-t + cp + €), (14) where <r differs but little from the angular velocity a> of the earth's rotation. These include the 'lunar diurnal' (o-=13 0, 943), the 'solar diurnal' (o- = 14°'959), and the 'luni- solar diurnal' (o-=<o = 15° '041 ). Lastly, we have the semi-diurnal tides, for which C=H'"sin 2 0.cos(<Tt + 2<p + e\ (15)t where a differs but little from 2o>. These include the 'lunar semi-diurnal' (o- = 28°*984), the 'solar semi-diurnal' (<r=30°), and the 'luni-solar semi-diurnal' (<r=2<o=30 0, 082). For a complete enumeration of the more important partial tides, and for the values of the coefficients H', H", H'" in the several cases, we must refer to the investigations of Darwin, already cited. In the Harmonic Analysis of Tidal Observations, which is the special object of these investigations, the only result of dynamical theory which is made use of is the general principle that the tidal elevation at any place must be equal to the sum of a series of simple-harmonic functions of the time, whose periods are the same as those of the several terms in the development of the disturbing potential, and are therefore known a priori. The amplitudes and phases of the various partial tides, for any particular port, are then determined by comparison with tidal observations extending over a * Eeference may be made to Laplace, Mecanique Celeste, Livre 13 me , Art. 2. The more complete development which has served as the basis of all recent accurate tidal work is due to Darwin, and is reprinted in his Papers, i. This development is only quasi-harmonic, certain elements which are only slowly variable being treated as constants, but adjustable from time to time. A strict harmonic development has recently been carried out by Doodson, Proc. Roy. Soc. A, c. 305 (1921). f It is evident that over a small area, near the poles, which may be treated as sensibly plane, the formulae (14) and (15) make fee r cos (<rt + <f> + e) , and f oc r 2 cos (<rt + 2$ + e), respectively, where r, w are plane polar co-ordinates. These forms have been used by anticipation in Arts. 211, 212. 362 Appendix to Chapter VIII sufficiently long period*. We thus obtain a practically complete expression which can be used for the systematic prediction of the tides at the port in question. f. One point of special interest in the Harmonic Analysis is the determination of the long-period tides. It has been already stated that under the influence of dissipative forces these must tend to approximate more or less closely to their equilibrium values. In the case of an ocean covering the globe it is at least doubtful whether the dissipative forces would be sufficient to produce an appreciable effect in the direction indicated. The amplitudes might therefore be expected to fall below those given by the equilibrium theory, for the dynamical reason explained in Arts. 206, 214. In the actual ocean, on the other hand, this consideration does not apply t, whilst the influence of friction is much greater. We may assume, then, that if the earth were absolutely rigid the long-period tides would have their full equilibrium values. As a matter of fact the lunar fortnightly, which is the only one whose amplitude can be inferred with any certainty from the observations, appears to fall short by about one-third. The discrepancy is attributed to elastic yielding of the solid body of the earth to the tidal distorting forces exerted by the moon. * It is of interest to note, in connection with Art. 187, that the tide-gauges, being situated in relatively shallow water, are sensibly affected by certain tides of the second order, which therefore have to be taken account of in the general scheme of Harmonic Analysis. t See the paper by Eayleigh cited on p. 355 ante. CHAPTEK IX SURFACE WAVES 227. We have now to investigate, as far as possible, the laws of wave- motion in liquids when the vertical acceleration is no longer neglected. The most important case not covered by the preceding theory is that of waves on relatively deep water, where, as will be seen, the agitation rapidly diminishes in amplitude as we pass downwards from the surface ; but it will be understood that there is a continuous transition to the state of things investigated in the preceding chapter, where the horizontal motion of the fluid was sensibly the same from top to bottom. We begin with the oscillations of a horizontal sheet of water, and we will confine ourselves in the first instance to cases where the motion is in two dimensions, of which one (x) is horizontal, and the other (y) vertical. The elevations and depressions of the free surface will then present the appearance of a series of parallel straight ridges and furrows, perpendicular to the plane xy. The motion, being assumed to have been generated originally from rest by the action of ordinary forces, will necessarily be irrotational, and the velocity-potential <f> will satisfy the equation dx* + dy 2 ' w with the condition ^- = (2) at a fixed boundary. To find the condition which must be satisfied at the free surface (p = const.), let the origin be taken at the undisturbed level, and let Oy be drawn vertically upwards. The motion being assumed to be infinitely small, we find, putting 12 = gy in the formula (4) of Art. 20, and neglecting the square of the velocity (q), J-8r»t>w (3) Hence if 77 denote the elevation of the surface at time t above the point (x, 0), we shall have, since the pressure there is uniform, .(4) provided the function F(t), and the additive constant, be supposed merged in the value of d<f>/dt. Subject to an error of the order already neglected, 364 Surface Waves [chap, ix this may be written <-JEL ; ; <S) Since the normal to the free surface makes an infinitely small angle (drj/dx) with the vertical, the condition that the normal component of the fluid velocity at the free surface must be equal to the normal velocity of the surface itself gives, with sufficient approximation, .(6) drj d(f> dt \_dy }, This is in fact what the general surface condition (Art. 9 (3)) becomes, if we put F(x. y, z y t)=y — r) y and neglect small quantities of the second order. Eliminating tj between (5) and (6), we obtain the condition dt* +9 dy U ' {) to be satisfied when y — 0. This is equivalent to Dp/Dt = 0. In the case of simple-harmonic motion, the time-factor being e i{(Tt+e) , this condition becomes ■"+-'! < 8 > 228. Let us apply this to the free oscillations of a sheet of water, or a straight canal, of uniform depth h, and let us suppose for the present that there are no limits to the fluid in the direction of x, the fixed boundaries, if any, being vertical planes parallel to xy. Since the conditions are uniform in respect to x, the simplest supposition we can make is that <f> is a simple-harmonic function of x ; the most general case consistent with the above assumptions can be derived from this by superposition, in virtue of Fourier's Theorem. We assume then </>=P cos kx.e^ t+t \ (1) where P is a function of y only. The equation (1) of Art. 227 gives 4- ¥P ^' (2) whence P = Ae^ + Be~ k y (3) The condition of no vertical motion at the bottom is dfyjdy — for y — — h, whence Ae- kh = Be kh , = %C, say. This leads to <£= Ccosh k(y + A,)cos lex .e ii<Tt+e) (4) The value of o- is then determined by Art. 227 (8), which gives a 2 = gk tanh kh (5) 227-228] Standing Waves 365 Substituting from (4) in Art. 227 (5), we find 7] = — cosh kh cos kx . e i{ot+t) t (6) if or, writing a = . cosh kh, i/ and retaining only the real part of the expression, 7j — a cos kx . sin (at + e) (7) This represents a system of 'standing waves,' of wave-length \— 2irjk, and vertical amplitude a. The relation between the period (2ir/<r) and the wave-length is given by (5). Some numerical examples of this dependence are given on p. 369. In terms of a we have ga cosh k (y + h) . , ^ x /ftX ^-.^yr 00 ^- 008 ^^) < 8 > and it is easily seen from Art. 62 that the corresponding value of the stream- function is qa sinh k (?/ + h) . , , M x If x, y be the co-ordinates of a particle relative to its mean position (x, y), we have *?-_l!4 <ty__d± nm d* 8^' ^ 9y' V ; if we neglect the differences between the component velocities at the points (#, i/) and (^ + x, y 4- y), as being small quantities of the second order. Sub- stituting from (8), and integrating with respect to t, we find cosh k(y + h) . 1 . , . , x N x = — a . ., . — - sin kx . sin (at + e) sinh kh $\vfak(y + K) 7 . . . , >. y = a . . ,, — - 7 cos kx . sin (at + e), sinh kh where a slight reduction has been effected by means of (5). The motion of each particle is rectilinear, and simple-harmonic, the direction of motion varying from vertical, beneath the crests and hollows (kx = m7r), to horizontal, beneath the nodes (kx = (m + -|) if). As we pass downwards from the surface to the bottom the amplitude of the vertical motion diminishes from <xcos&# to 0, whilst that of the horizontal motion diminishes in the ratio cosh kh : 1. When the wave-length is very small compared with the depth, kh is large, and therefore tanh kh — 1*. The formulae (11) then reduce to x = — aepy sin kx . sin (at + e), y = ae ky cos kx . sin (at + e), . . .(12) with a 2 = gk (13) * This case may of course be more easily investigated independently. ,(ii) 366 Surface Waves [chap. IX The motion now diminishes rapidly from the surface downwards ; uhus at a depth of a wave-length the diminution of amplitude is in the ratio e~ 2n or 1/535. The forms of the lines of (oscillatory) motion (yjr = const.), for this case, are shewn in the annexed figure. In the above investigation the fluid is supposed to extend to infinity in the direction of x, and there is consequently no restriction to the value of k. The formulae also give, however, the longitudinal oscillations in a canal of finite length, provided k have the proper values. If the fluid be bounded by the vertical planes x = 0, x — l (say), the condition d<j>/dx — is satisfied at both ends provided sinkl = 0, or kl^mir, where ra = l, 2, 3, The wave-lengths of the normal modes are therefore given by the formula X = 2Z/m. Cf. Art. 178. 229. The investigation of the preceding Art. relates to the case of 'standing' waves; it naturally claimed the first place, as a straightforward application of the usual method of ascertaining the normal modes of oscilla- tion of a system about a state of equilibrium. In the case, however, of a sheet of water, or a canal, of uniform depth, extending horizontally to infinity in both directions, we can, by super- position of two systems of standing waves of the same wave-length, obtain a system of progressive waves which advance unchanged with constant velocity. For this, it is necessary that the crests and troughs of one component system should coincide (horizontally) with the nodes of the other, that the amplitudes of the two systems should be equal, and that their phases should differ by a quarter-period. Thus if we put v—Vi±V2> (1) where rjx = a sin kx cos at, r) 2 = a cos kx sin at, (2) we get 7} = a sin (kx ± at), (3) which represents an infinite train of waves travelling in the negative or positive direction of x, respectively, with the velocity c given by c=^ = (|tanhM)*, (4) where the value of a has been substituted from Art. 228 (5). In terms of 228-229] Progressive Waves 367 the wave-length (X) we have tanh^)* (5) V2tt X / When the wave-length is anything less than double the depth, we have tanh kh — 1, sensibly, and therefore* 'm' ( 6) -(DM' ,2tt; On the other hand when X is moderately large compared with h we have tanhM = M, nearly, so that the velocity is independent of the wave-length, being given by c = (gh)i (7) as in Art. 170. This result is here obtained on the assumption that the wave-profile is a curve of sines, but Fourier's Theorem shews that the restriction is now to a great extent unnecessary. It appears, on tracing the curve y = (tanh^)/a; > or from a numerical table to be given presently, that for a given depth h the wave-velocity increases constantly with the wave-length, from zero to the asymptotic value (7). Let us now fix our attention, for definiteness, on a train of simple-harmonic waves travelling in the positive direction, i.e. we take the lower sign in (1) and (3). It appears, on comparison with Art. 228 (7), that the value of rji is deduced by putting e = \tt, and subtracting \tt from the value of facf, and that of 772 by putting e = 0, simply. This proves a statement made above as to the relation between the component systems of standing waves, and also enables us to write down at once the proper modifications of the remaining formulae of the preceding Art. Thus, we find, for the component displacements of a particle, cosh k (y + h) n \ x = Xl -x 2 = a s . nh ^ ' coB(fo-<rQ, sinh k(y + h) . /7 jX This shews that the motion of each particle is elliptic-harmonic, the period (27r/cr, = X/c) being that in which the disturbance travels over a wave-length. The semi-axes, horizontal and vertical, of the elliptic orbits are cosh k (y + h) . sinh k(y + h) a . ^ yT — and a . . , 7 — - , smh kh , sinn kh respectively. These both diminish from the surface to the bottom {y = — h), where the latter vanishes. The distance between the foci is the same for all * Green, "Note on the Motion of Waves in Canals," Camb. Trans, vii. (1839) [Papers, p. 279]. + This is merely equivalent to a change of the origin from which x is measured. .(8) 368 Surface Waves [chap. IX the ellipses, being equal to a cosech kh. It easily appears, on comparison of (8) with (3), that a surface-particle is moving in the direction of wave- propagation when it is at a crest, and in the opposite direction when it is in a trough*. When the depth exceeds half a wave-length, e~ kh is very small, and the formulae (8) reduce to x = ae ky cos (Jcx— o-t), ■y = ae k ysin(k% — <rt), (9) so that each particle describes a circle, with constant angular velocity a, = (27r<7/\)£-j\ The radii of these circles are given by the formula ae ky , and therefore diminish rapidly downwards. In the table given below, the second column gives the values of sech kh corresponding to various values of the ratio hjX. This quantity measures the ratio of the horizontal motion at the bottom to that at the surface. The third column gives the ratio of the vertical to the horizontal diameter of the elliptic orbit of a surface-particle. The fourth and fifth columns give the ratios of the wave-velocity to that of waves of the same length on water of infinite depth, and to that of 'long' waves on water of the actual depth, respectively. The tables of absolute values of periods and wave-velocities, on the opposite page, are abridged from Airy's treatise J. The value of g adopted by him is 32*16 ft./sec. 2 The possibility of progressive waves advancing with unchanged form is limited, theo- retically, to the case of uniform depth ; but the numerical results shew that a variation in the depth will have no appreciable influence, provided the depth everywhere exceeds (say) half the wave-length. h/\ sech kh tanh kh cl(gk-rf cl(gh)* o-oo 1-000 o-ooo o-ooo 1-000 •01 •998 •063 •250 •999 •02 •992 •125 •354 •997 •03 •983 •186 •432 •994 •04 •969 •246 •496 •990 •05 •953 •304 •552 •984 •06 •933 •360 •600 •977 •07 •911 •413 •643 •970 •08 •886 •464 •681 •961 •09 •859 •512 •715 •951 •10 •831 •557 •746 •941 •20 •527 •850 •922 •823 •30 •297 •955 •977 •712 •40 •161 •987 •993 •627 •50 •086 •996 •998 •563 •60 •046 •999 •999 •515 •70 •025 1-000 1-000 •477 •80 •013 1-000 1-000 •446 •90 •007 1-000 1-000 •421 1-00 •004 1-000 1-000 •399 00 •000 1-000 1-000 •000 * The results of Arts. 228, 229, for the case of finite depth, were given, substantially, by Airy, "Tides and Waves," Arts. 160... (1845). t Green, I.e. % "Tides and Waves," Arts. 169, 170. 229-230] Numerical Results 369 Length of wave, in feet Depth of water, in feet 1 10 100 j 1000 | 10,000 | 1 0*442 1-873 17-645 176-33 1763-3 10 0-442 1-398 5-923 55-80 557-62 100 0-442 1-398 4-420 18-73 176*45 1000 0-442 1-398 4-420 13-98 59-23 10,000 0-442 1-398 4-420 13-98 44-20 Depth of Length of wave, in feet water, in feet 1 10 100 1000 | 10,000 1 2-262 5-339 5-667 5-671 5-671 5-671 10 2-262 7-154 16-88 17-92 17-93 17-93 100 2-262 7-154 22-62 53-39 56-67 56-71 1000 2-262 7-154 22-62 71-54 168-8 1.79-3 10,000 2-262 7-154 22-62 71-54 226-2 567-1 We remark, finally, that the theory of progressive waves may be obtained, without the intermediary of standing waves, by assuming at once, in place of Art. 228(1), Q = J> e i(<rt-kX) ( 10 ) The conditions to be satisfied by P are exactly the same as before, and we easily find, in real form, 7] — a sin (Jex — at), (11) qa cosh k (y + h) /7 . __. with the same determination of a as before. From (12) all the preceding results as to the motion of the individual particles can be inferred without difficulty. 230. The energy of a system of standing waves of the simple-harmonic type is easily found. If we imagine two vertical planes to be drawn at unit distance apart, parallel to xy, the potential energy per wave-length of the fluid between these planes is \gp rpdx. Jo Substituting the value of w from Art. 228 (7), we obtain lgpa 2 \. sin 2 (at + e) (1) 370 Surface Waves [chap, ix The kinetic energy is, by the formula (1) of Art. 61, dx. *'£[+' fyjy=o Substituting from Art. 228 (8), and remembering the relation between cr and k, we obtain lgpa 2 \. cos 2 (at + e) (2) The total energy, being the sum of (1) and (2), is constant, and equal to \gpa 2 \. We may express this by saying that the total energy per unit area of the water-surface is \gpci 2 . A similar calculation may be made for the case of progressive waves, or we may apply the more general argument explained in Art. 174. In either way we find that the energy at any instant is half potential and half kinetic, and that the total amount, per unit area, is ^gpa 2 . In other words, the energy of a progressive wave-system of amplitude a is equal to the work which would be required to raise a stratum of the fluid, of thickness a, through a height \a. 231. We next consider the oscillations of the common boundary of two superposed liquids which are otherwise unlimited. Taking the origin at the mean level of the interface we may write <l> = CePy cos Jcaetrt, </>' = C'e-^coskxe™ 1 , (1) where the accents relate to the upper fluid. For these satisfy Art. 227 (1) atid vanish for y = — oo and y = + oo , respectively. Hence if the equation of the disturbed surface is 7] = a cos kx e i<rt (2) we must have -kC = kC' = iaa (3) by Art. 227 (6). Again, the formulae p = K~ m 7 = K- gy (4) give p (iaC — ga) = p' (io-C - ga) (5) as the condition for continuity of pressure at the interface. Substituting the values of G and C" from (3) we have •"-«*'^ (6) The velocity of propagation of waves of length 27r/& is therefore given by C '=f.^: (7) fc p + p The presence of the upper fluid has therefore the effect of diminishing the velocity of propagation of waves of any given length in the ratio {(1 —8)/(l +s)}b, where s is the ratio of the density of the upper to that of 230-231] Superposed Fluids 371 the lower fluid. This diminution has a two-fold cause; the potential energy of a given deformation of the common surface is diminished in the ratio 1—5, whilst the inertia is increased in the ratio 1 + s *. As a numerical example, in the case of water over mercury (s -1 = 13'6) the wave- velocity is diminished in the ratio '929. It is to be noticed, in this and in other problems of the kind, that there is a discontinuity of motion at the common surface. The normal velocity (— d(j>/dy) is of course continuous, but the tangential velocity (— d(f>jdoc) changes sign as we cross the surface; in other words we have (Art. 151) a vortex-sheet. This is an extreme illustration of the remark, made in Art. 17, that the free oscillations of a liquid of variable density are not necessarily irrotational. In reality the discontinuity, if it could ever be originated, would be immediately abolished by viscosity, and the vortex-sheet replaced by a film of vorticityf. If p < //, the value of a is imaginary. The undisturbed equilibrium- arraugement is then unstable. If the two fluids are confined between rigid horizontal planes y = — h y y = h', we assume in place of (1) $ = C cosh k(y + h) cos kxe i<Tt , <p' = C cosh k(y — h') cos kx e i<rt , (8) since these make d<p/dy = 0, dcp'ldy=0 at the respective planes. Hence — kCsinh. kh=W sinh kh' = iaa (9) The continuity of pressure requires p (io~C cosh kh — ga)=p (icrC cosh kh! —go) (10) Eliminating C, C, g2= gk(p-p') ^ (n) p coth&A + p'coth kh' When kh and kh' are both very great this reduces to the form (6). When kh' is large and kh small we find C 2 = .2/ F= A_A^ (12) approximately, the main effect of the presence of the upper fluid being the change in the potential energy of a given deformation. Its kinetic energy is small compared with that of the lower fluid. * This explains why the natural periods of oscillation of the common surface of two liquids of very nearly equal density are very long compared with those of a free surface of similar extent. The fact was noticed by Benjamin Franklin in the case of oil over water; see a letter dated 1762 {Complete Works, London, n. d., ii. 142). Again, near the mouths of some of the Norwegian fiords there is a layer of fresh over salt water. Owing to the comparatively small potential energy involved in a given deformation of the common boundary, waves of considerable height in this boundary are easily produced. To this cause is ascribed the abnormal resistance occasionally experienced by ships in those waters. See Ekman, "On Dead- Water," Scientific Results of the Norwegian North Polar Expedition, pt. xv. Christiania, 1904. Beference may also be made to a paper by the author, "On Waves due to a Travelling Disturbance, with an application to Waves in Superposed Fluids," Phil. Mag. (6), xxxi. 386 (1916). t The solution, taking account of viscosity, is given by Harrison, Proc. Lond. Math. Soc. (2), vi. 396 (1908)- 372 Surface Waves [chap, ix When the upper surface of the upper fluid is free we may assume <p <= Ccosh 1c(y + h) cos kx e i(Tt , <p' = (A cosh ky + B sinh ky) cos kx e iat (13) The kinematical condition is then — kCs'\uhkh-= — B=io-a (14) The condition for continuity of pressure at the interface is p (io-C cosh kh — ga)=p (iaA—ga) (15) The condition for constancy of pressure at the free surface is given by Art. 227 (8) provided we put y = h! after the differentiations. Thus <r 2 {A cosh kh' + B sinh kh')=gk (A sinh kh! + B cosh kh') (16) The elimination of A, B, C between (14), (15), (16) leads to the equation o- 4 (p coth kh coth kh' + p) -<x 2 p (coth kh' + coth kh) gk -f (p -p ) g 2 k 2 =0 (17) Since this is a quadratic in a 2 , there are two possible systems of waves of any given period (27r/o-). This is as we should expect, for when the wave-length is prescribed the system has virtually two degrees of freedom, so that there are two independent modes of oscilla- tion about the state of equilibrium. For example, in the extreme case where p'/p is small, one mode consists mainly in an oscillation of the upper fluid which is almost the same as if the lower fluid were solidified, whilst the other mode may be described as an oscillation of the lower fluid which is almost the same as if its upper surface were free. The ratio of the amplitude at the upper to that at the lower surface is found to be fc a , 18) kc 2 cosh kh' — g sinh kh! * Of the various special cases that may be considered, the most interesting is that in which kh is large ; i.e. the depth of the lower fluid is great compared with the wave- length. Putting coth kh = l, we see that one root of (17) is now * 2 =gk, (19) exactly as in the case of a single fluid of infinite depth, and that the ratio of the ampli- tudes is e kh '. This is merely a particular case of a general result stated near the end of Art. 233 ; it will in fact be found on examination that there is now no slipping at the common boundary of the two fluids. The second root of (17) is, on the same supposition, <T" p-p p coth kh' + p and for this the ratio (18) assumes the value fa (20) -(t-i\ e -kh' (21) If in (20) and (21) we put kh' = cc , we fall back on a former case. If on the other hand we make kh' small, we find p-HD**- ( 22 > and the ratio of the amplitudes is (23) -a-')- These problems were first investigated by Stokes*. The case of any number of super posed strata of different densities has been treated by Webbt and Greenhill $ . * "On the Theory of Oscillatory Waves," Camb. Trans, viii. (1847) [Papers, i. 212]. t Math. Tripos Papers, 1884. X "Wave Motion in Hydrodynamics," Amer. Journ. of Math. ix. (1887). 231-232] Waves on a Surface of Discontinuity 378 232. Let us next suppose that we have two fluids of densities p, p', one beneath the other, moving parallel to x with velocities IT, U', respectively, the common surface (when undisturbed) being of course plane and horizontal. This is virtually a problem of small oscillations about a state of steady motion. We write, then, <£ = -£7# + <£j, tf^-U'x + ti, (1) where fa, (pi are by hypothesis small. The velocity of either fluid at the interface may be regarded as made up of the velocity of this surface itself, and the velocity of the fluid relative to it. Hence if rj be the ordinate of the displaced surface we have, considering vertical components, dt^ dx dy' dt* dx ~ dy' w as the kinematical conditions to be satisfied for y = 0. Again, the formula for the pressure in the lower fluid is ?-£-*{(*-£)"+(t)'}-»-- -%-<%-«+ - <■» the terms omitted being either of the second order, or irrelevant to the present purpose. Hence the condition of continuity of pressure is p(£+*3h*)-'(£ + "'£-") < 4 > We have seen, in various connections, that in oscillations about steady motion there is not necessarily uniformity of phase throughout the system, and in the present case it would not be found possible to satisfy the con- ditions on such an assumption. Assuming both fluids to be of unlimited depth, the appropriate course is to write famCf*****-**, fa' - C'f-*****-**, (5) and v **aJ<f*-**> (6) The conditions (2) then give i(<r-kU)a = -W, i(<r-kU')a = kC' t (7) whilst, from (4), p{i(* -kU)C '- ga}= p' '{i(tr -leU') C - ga) (8) Hence p(<r -kU? + p (a- -rkU'f = gk(p - p) (9) or .(10) T pU+p U' (g p-p pp -p. r> . s *- p+p' ± Wp + p ' (p + P 'f (U U) The first term on the right-hand side may be called the mean velocity of the * These are particular cases of the general boundary-condition (3) of Art. 9, as is seen by writing F=y-r), and neglecting small terms of the second order. 374 Surface Waves [chap, ix two currents. Relatively to this there are waves travelling with velocities ± c, given by c2 = c o 2 -r^Y 2 (^- ^') 2 , (ii) KP + P) where cq denotes the wave- velocity in the absence of currents (Art. 231). It is to be noticed however that the values of <r given by (9) are imaginary if (U-Uy>{. p2 -^ (12) K pp The common boundary is therefore unstable for sufficiently small wave- lengths. This result would indicate that, if there were no modifying cir- cumstances, the slightest breath of wind would ruffle the surface of water. A more complete investigation will be given later, taking account of capillary forces, which act in the direction of stability. If p = p', or if g — 0, the plane form of the surface is (on the present reckoning) unstable for all wave-lengths. This result illustrates the statement, as to the instability of surfaces of dis- continuity in a liquid, made in Art. 79*. The case of p = p\ with U= U', is of some interest, as illustrating the flapping of sails and flags f. We may conveniently simplify the question by putting U— U' = 0; any common velocity may be superposed afterwards if desired. On these suppositions the equation (8) reduces to <r 2 = 0. On account of the double root the solution has to be completed by the method explained in books on Differential Equations. In this way we obtain the two independent solutions v = ae ik *, ^ = 0, </>/ = (), (13) and v = ate ikx , ^ = -^.e te , fa' = %-^.e ikx (14) The former solution represents a state of equilibrium; the latter gives a system of stationary waves with amplitude increasing proportionally to the time. In this form of the problem there is no physical surface of separation to begin with ; but if a slight discontinuity of motion be artificially produced, e.g. by impulses applied to a thin membrane which is afterwards dissolved, the discontinuity will persist, and, as we have seen, the height of the corrugations will continually increase. An interesting application of the same method is to the case of a jet of thickness 2b moving through still fluid of the same density J. Taking the origin in the medial plane we write, for the disturbed jet <£= — Ux + cf) , and for the fluid on the two sides <£ = 0i for y > b, and $ = <£ 2 for y < - b. We also denote by r) X , rj 2 the normal displacements of the two surfaces y = b and y= — 6, respectively. The proper assumptions are then cf> 1 ^A 1 e- k ye i i (Tt - kx \ <t) 2 = A 2 e k ve i ( (Tt - kx ), } to-OiW-m, 77 2 =<7 2 e^-**), I (15) S = ( A cosh ky + B sinh hy) e<(**-**) . J * This instability was first remarked by Helmholtz, I.e. ante p. 22. t Rayleigh, Proc. Lond. Math. Soc. (1) x. 4 (1879) [Papers i. 361]. J Rayleigh I.e. 232-233] Instability of Jets 375 There are obviously two types of disturbance, in which r} X = r) 2i and rj 1 = -jj 2 , respectively. In the former case we have Ci = C 2 , A =0, A 2 = — A x . The kinematical conditions (2) at the surface y = b then give iaC^kA^-™, i((r-kU)C=-kB coshkh, (16) whilst the continuity of pressure requires, gravity being omitted, (o- - kU) B $inhkk=<rA 1 e- kh (17) Hence (o--££0 2 tanhM+o- 2 =0 (18) If the thickness 2b is small compared with the wave-length of the disturbance, we have <r= ±ikU s l{kh\ (19) approximately, indicating a very gradual instability, as is often observed in the case of filaments of smoke. In the case of symmetry (^ = — rj 2 ), we should find (<r-kU) 2 cothkh + <T*=0 (20) in place of (18). The theory of progressive waves may also be investigated, in a very compact manner, by the method of Art. 175*. Thus if <f>, -v/r be the velocity- and stream -functions when the problem has been reduced to one of steady motion, we assume r ?X = _ (# + iy) + i ae ih(x+iy) + ip e -ik(x+iy) } C whence ® = — x — {air 1 ® — fie 1 ®) sin kx, 1 a) T = - y + (aer 7 ® H- /3e k v) cos ky. This represents a motion which is periodic in respect to x, superposed on a uniform current of velocity c. We assume that ka and kft are small quantities; in other words, that the amplitude of the disturbance is small compared with the wave-length. The profile of the free surface must be a stream-line; we take it to be the line i/r= 0. Its form is then given by (1), viz. to a first approximation we have y = (a + /3) cos kx, (2) shewing that the origin is at the mean level of the surface. Again, at the bottom (y= —h) we must also have ty = const. ; this requires ae kh + Pe~ kh = 0. The equations (1) may therefore be put in the forms — = — x + C cosh k(y + h) sin kx, c * Kayleigh, I.e. ante p. 260. y + C sinh k(y + h) cos kx. c .(3) 376 Surface Waves [chap, ix The formula for the pressure is ?-— »-»{S)'+d)l c 2 = const. — gy — -= {1 — 2&0 cosh k(y + h) cos &#}, if we neglect k 2 C 2 . Since the equation to the stream-line ifr = is y = (7 sinh Ich cos A;a?, (4) approximately, we have, along this line, - = const. 4- (&c 2 coth Ich — g) y. P The condition for a free surface is therefore satisfied, provided 9 , tanh hh /KX c= ^-^wr" (5) This determines the wave-length (2ir/k) of possible stationary undulations on a stream of given uniform depth h, and velocity c. It is easily seen that the value of hh is real or imaginary according as c is less or greater than (ghft. If we impress on everything the velocity — c parallel to x, we get progressive waves on still water, and (5) is then the formula for the wave- velocity, as in Art. 229. When the ratio of the depth to the wave-length is sufficiently great, the formulae (1) become ®== — x + fie 1 ® sin kx, — = — y + fie 1 ® cos kx, (6) leading to £ = const. - gy - % {1 - 2k fie 1 ® cos kx + k 2 fi*e 21 ®} (7) If we neglect k 2 fi 2 y the latter equation may be written - = const. + (kc 2 — g)y + kcyjr (8) r Hence if c 2 = g/k, (9) the pressure will be uniform not only at the upper surface, but along every stream-line -ty = const.* This point is of some importance; for it shews that the solution expressed by (6) and (9) can be extended to the case of any number of liquids of different densities, arranged one over the other in horizontal strata, provided the uppermost surface be free, and the total depth infinite. And, since there is no limitation to the thinness of the strata, we may even include the case of a heterogeneous liquid whose density varies continuously with the depth. Cf. Art. 235. * This conclusion, it must be noted, is limited to the case of infinite depth. It was first remarked by Poisson, I.e. post p. 384. .(11) 233-234] Artifice of Steady Motions 377 Again, to find the velocity of propagation of waves over the common horizontal boundary of two masses of fluid which are otherwise unlimited, we may assume ^= -y+/3e^cos£#, ^-=-y + 0e-*ycos&F, (10) c c where the accent relates to the upper fluid. For these satisfy the condition of irrotational motion, V 2 ^ = ; and they give a uniform velocity c at a great distance above and below the common surface, at which we have \^ = ■*//■', =0, say, and therefore y = /3cos&#, approxi- mately. The pressure-equations are 7) (P 1 *- = const, —gy — ^ (1 - 2k(3e kv cos Tcx\ P * v' c 2 S = const. - gy - - (1 + 2^e~ k v cos kx), p ' 2, i which give, at the common surface, — = const. — (g — kc 2 )y, -, = const. — (g+kc 2 )y, (12) the usual approximations being made. The condition p=p' thus leads to M-^'. ( 13 ) as in Art. 231. 234. As a further example of the method we take the case of two super- posed currents, already treated by the direct method in Art. 232. The fluids being unlimited vertically, we assume yjr= - U{y- fie** cos kx}, yjr' =- U'{y-0er*v coa kx], (1) for the lower and upper fluids respectively. The origin is taken at the mean level of the common surface, which is assumed to be stationary, and to have the form y = /3 cos kx (2) The pressure-equations give - = const. - gy — \ U 2 (1 — 2k fie^ cos kx), Pj = const. - gy -%U' 2 (1 + 2k/3er* cos kx), P whence, at the common surface, (3) 2 = const. + (kU 2 -g)y, ^_ = const. ~(kU' 2 +g)y (4) Since we must have p — p f over this surface, we get P U* + p'U'*=l(p-p') (5) This is the condition for stationary waves on the common surface of the two currents U, U'. It may be written ( pU + p'Uy _ g_ p-l _ 9 p> \ p+p' ) k-p+p' (p+p') i(u U)> — w which is, easily seen to be equivalent to Art. 232 (10). 378 Surface Waves [chap, ix When the currents are confined by fixed horizontal planes y= — h, y = h', we assume (7) The condition for stationary waves on the common surface is then found to be pU 2 cothkh+p'U ,2 cothkh'=£(p-p') (8)* 235. The theory of waves in a heterogeneous liquid may be noticed, foi the sake of comparison with the case of homogeneity. The equilibrium value p of the density will be a function of the vertical co-ordinat( (y) only. Hence, writing P**Po+P'> P=po+p\ (1) where p . is the equilibrium pressure, the equations of motion, viz. du dp dv dp nr-£> "s--^-«- (2) fc+-£+'|-* m , du dp' dv dp' , become <">Tt = -£' »%--■%-*>> (4) !+»lr ' <*> small quantities of the second order being omitted. The fluid being incompressible, the equation of continuity retains the form du dv a^ + ^= ' < 6 > so that we may write ■— *• *-s < 7 > u— - - "- — Eliminating jo' and p' we find t fft'«-»S»- ■ At a free surface we must have Dp/Dt=0, or !—%-•* < 9 > Hence, and from (4), we must have 8^ 9ty ^ = *aJ ( 10 > at such a surface. To investigate cases of wave-motion we assume that ^ oc e i(fft-Jex) *jj\ The equation (8) becomes 3-**£*«-s*)- » whilst the condition (10) takes the form ^"^^ = ° ( 13 ) * Greenhill, Z.c. ante p. 372. f Cf. Love, "Wave Motion in a Heterogeneous Heavy Liquid," Proc. Lond. Math. Soc. xxii. 307 (1891). ■"♦-'&-S)*-° (16) 234-235] Waves in Heterogeneous Liquid 379 These are satisfied, whatever the vertical distribution of density, by the assumption that \ls varies as e ky . provided o*=gk (14) For a fluid of infinite depth the relation between wave-length and period is then the same as in the case of homogeneity (cf. Art. 229), and the motion is irrotational. For further investigations it is necessary to make some assumption as to the relation between p and y. The simplest is that **«-*, (15) in which case (12) takes the form w The solution is ^(i^+^e^-^, (17) where X 1} X 2 are the roots of A*-j8A + (2f-l\iP«0 (18) We first apply this to the oscillations of liquid filling a closed rectangular vessel*. The quantity k may be any multiple of w/l, where 7 denotes the length. If the equations to the horizontal boundaries be y =0, y = h, the condition d\jr/dx=0 gives A+B = 0, Ae\ h +Be** h =0, (19) whence e(\-^)^ = l, or Xj - X 2 = 2iW/A, (20) where 5 is integral. Hence, from (18), # X 1 = l i 3-f isir/h, X 2 =J/3-M7r/A, (21) and therefore (^-i) ^ 2=x i X 2 = ¥/3 2 + ^ (22) We verify that o- is real or imaginary, i.e. the equilibrium arrangement is stable or unstable, according as )3 is positive or negative, i.e. according as the density diminishes or increases upwards t. The case where the fluid (of depth h) has a free surface may serve as an illustration of the theory of 'temperature seiches' in lakes J. Assuming the roots of (18) to be complex, say X = |/3±ira, (23) with m 2 = (^f-l N ) ^ 2 -^, (24) 2 -i 4 » we have y\r = Ce^ v sin my, (25) the origin of y being taken at the bottom. The surface-condition (13) gives ^/3sinraA + mcosmA=^-2 sin mh (26) With the help of (24) this may be written *»«*-». ggqjpsSiPp . < 27) * Kayleigh, "Investigation of the Character of the Equilibrium of an Incompressible Heavy Liquid of Variable Density," Proc. Lond. Math. Soc. (1) xiv. 170 [Papers, ii. 200]. Eeference may also be made to a paper by the author "On Atmospheric Oscillations," Proc. Boy. Soc. lxxxiv. 566, 571 (1910), where another law of density is considered. t The case of waves on a liquid of finite depth is discussed by Love (I.e.). See also Burnside, "On the Small Wave-Motions of a Heterogeneous Fluid under Gravity," Proc. Lond. Math. Soc. (1) xx. 392 (1889). t Discussed by Wedderburn, Trans. R. S. Edin. xlvii. 619 (1910) and xlviii. 629 (1912). 380 Surface Waves [chap, ix from which the values of mh are to be found. They are given graphically by the inter- sections of the curves y = tan.z, V=Jj^v (28) where n=(3h, a 2 = k 2 h 2 — \$ 2 h 2 . — The only case of interest is when $h is small. We have, then, mh = S7r, approximately, and thence h 2 h 2 ••-*•??+*»■ (29) which is seen to be identical with (22) when the square of /3A is neglected. It appears in fact from (25) that the vertical motion at the free surface is very slight. The maximum vertical disturbance is at the levels # = (* — £) it. When the roots of (18) are real we should get only a slight correction to the formula (T 2 =gk tanh hh which holds for a homogeneous fluid. 236. The investigations of Arts. 227-234 relate to a special type of waves; the profile is simple-harmonic, and the train extends to infinity in both directions. But since all our equations are linear (so long as we confine ourselves to a first approximation), we can, with the help of Fourier's Theorem, build up by superposition a solution which shall represent the effect of arbitrary initial conditions. Since the subsequent motion is in general made up of systems of waves, of all possible lengths, travelling in either direction, each with the velocity proper to its own wave-length, the form of the free surface will continually alter. The only exception is when the wave-length of every system which is present in sensible amplitude is large compared with the depth of the fluid. The velocity of propagation, viz. \!{gh), is then independent of the wave-length, so that in the case of waves travelling in one direction only, the wave-profile remains unchanged in form as it advances (Art. 170). The effect of a local disturbance of the surface, in the case of infinite depth, will be considered presently; but it is convenient to introduce in the first place the very important conception of 'group- velocity,' which has application, not only to water-waves, but to every case of wave-motion where the velocity of propagation of a simple-harmonic train varies with the wave-length. It has often been noticed that when an isolated group of waves, of sensibly the same length, is advancing over relatively deep water, the velocity of the group as a whole is less than that of the individual waves composing it. If attention be fixed on a particular wave, it is seen to advance through the group, gradually dying out as it approaches the front, whilst its former place in the group is occupied in succession by other waves which have come forward from the rear*. The simplest analytical representation of such a group is obtained by the superposition of two systems of waves of the same amplitude, and of nearly * Scott Kussell, "Keport on Waves," Brit. Ass. Rep. 1844, p. 369. There is an interesting letter on this point from W. Froude, printed in Stokes' Scientific Correspondence, Cambridge, 1907, ii. 156. 235-236] Group Velocity 381 but not quite the same wave-length. The corresponding equation of the free surface will be of the form 7) = a sin (kx — <rt) + a sin (k f x— at) = 2acos{i(k-k')x-i(<T-<T')t\sm{%(Jc + k')x-l(<T + <T')t}. ...(1) If k, k' be very nearly equal, the cosine in this expression varies very slowly with x ; so that the wave-profile at any instant has the form of a curve of sines in which the amplitude alternates gradually between the values and 2a. The surface therefore presents the appearance of a series of groups of waves, separated at equal intervals by bands of nearly smooth water. The motion of each group is then sensibly independent of the presence of the others. Since the distance between the centres of two successive groups is 27r/(A?— k'), and the time occupied by the system in shifting through this space is 2ir/(a — a'), the group- velocity ( U, say) is = (a — <r')/(k — k'), or *-£. ™ ultimately. In terms of the wave-length X (= 27r/k), we have . ^r-**. (3) where c is the wave -velocity. This result holds for any case of waves travelling through a uniform medium. In the present application we have = (|tanhM) , (4) ity, , /, 2kh \ and therefore, for the group- velocity, d(kc )_ , (^ , 2kh dk The ratio which this bears to the wave-velocity c increases as kh diminishes, being \ when the depth is very great, and unity when it is very small, compared with the wave-length. The above explanation seems to have been first given by Stokes*. The extension to a more general type of group was made by Rayleighf and GouyJ. Another derivation of (3) can be given which is, perhaps, more intuitive. In a medium such as we are considering, where the wave-velocity varies with the frequency, a limited initial disturbance gives rise in general to a wave- system in which the different wave-lengths, travelling with different velocities, * Smith's Prize Examination, 1876 [Papers, v. 362]. See also Bayleigh, Theory of Sound, Art. 191. t Nature, xxv. 52 (1881) [Papers, i. 540]. % " Sur la vitesse de la lumiere," Ann. de Chim. et de Phys. xvi. 262 (1889). It has recently been pointed out that the theory had been to some extent anticipated by Hamilton, working from the optical point of view, in 1839; see Havelock, Cambridge Tracts, No. 17 (1914), p. 6. 382 Surface Waves [chap. IX are gradually sorted out (Arts. 238, 239). If we regard the wave-length \ as a function of oc and t, we have 9X rr^ dt doc o, .(6) since X, does not vary in the neighbourhood of a geometrical point travelling with velocity U; this is, in fact, the definition of U. Again, if we imagine another geometrical point to travel with the waves, we have d\ dX _ dc _ dc dX dt doc doc dXdoc •(7) the second member expressing the rate at which two consecutive wave-crests are separating from one another. Combining (6) and (7), we are led, again, to the formula (3)*. This formula admits of a simple geometrical representation t. If a curve be con- structed with X as abscissa and c as ordinate, the group-velocity will be represented by N the intercept made by the tangent on the axis of c. Thus, in the figure, PN represents the wave-velocity for the wave-length ON, and OT represents the group-velocity. The frequency of vibration, it may be noticed, is represented by the tangent of the angle PON. In the case of gravity- waves on deep water, c oc \z ; the curve has the form of the parabola y 2 = 4a#, and OT=\PN, i.e. the group-velocity is one-half the wave-velocity. 237. The group-velocity has moreover a dynamical, as well as a geo- metrical, significance. This was first shown by Osborne Reynolds J, in the case of deep-water waves, by a calculation of the energy propagated across a * See a paper "On Group- Velocity," Proc. Lond. Math. Soc. (2) i. 473 (1904). The subject is further discussed by G. Green, "On Group- Velocity, and on the Propagation of Waves in a Dispersive Medium," Proc. R. S. Edin. xxix. 445 (1909). t Manch. Mem. xliv. No. 6 (1900). J "On the Rate of Progression of Groups of Waves, and the Rate at which Energy is Transmitted by Waves," Nature, xvi. 343 (1877) [Papers, i. 198]. Reynolds also constructed a model which exhibits in a very striking manner the distinction between wave-velocity and group. velocity in the case of the transverse oscillations of a row of equal pendulums whose bobs are connected by a string. 236-237] Transmission of Energy 383 vertical plane. In the case of infinite depth, the velocity-potential corre- sponding to a simple-harmonic train 7) = a sin k (x — ct) (8) is <p = ac $ y cos k (x — ct), (9) as may be verified by the consideration that for y = we must have drj/dt = — d(j)/dy. The variable part of the pressure is p dcf)/dt, if we neglect terms of the second order. The rate at which work is being done on the fluid to the right of the plane x is therefore — I P^r-dy ' — pa 2 k 2 c z sin 2 k (x — ct) I e Uy dy J — oo OX J — co = \gpa 2 c sin 2 k (x — ct), (10) since c 2 = g/k. The mean value of this expression is \gpa 2 c. It appears on reference to Art. 230 that this is exactly one-half of the energy of the waves which cross the plane in question per unit time. Hence in the case of an isolated group the supply of energy is sufficient only if the group advance with half the velocity of the individual waves. It is readily proved in the same manner that in the case of a finite depth h the average energy transmitted per unit time is * *^( 1+ bSh)' (11) which is, by (5), the same as ifl*»'x*g> (12) Hence the rate of transmission of energy is equal to the group-velocity, d (kc)/dk, found independently by the former line of argument. This identification of the kinematical group- velocity of the preceding Art. with the rate of transmission of energy may be extended to all kinds of waves. It follows indeed from the theory of interference groups (p. 381), which is of a general character. For let P be the centre of one of these groups, Q that of the quiescent region next in advance of P. In a time r which extends over a number of periods, but is short compared with the time of transit of a group, the centre of the group will have moved to P', such that PP' = Ur, and the space between P and Q will have gained energy to a corresponding amount. Another investigation, not involving the notion of 'interference,' was given by Rayleigh (I.e.). From a physical point of view the group-velocity is perhaps even more important and significant than the wave-velocity. The latter may be greater or less than the former, and it is even possible to imagine mechanical media in which it would have the opposite direction ; i.e. a disturbance might be * Rayleigh, "On Progressive Waves," Proc. Lond. Math. Soc. (1) ix. 21 (1877) [Papers, i. 322]; Theory of Sound, i. Appendix. 384 Surface Waves [chap, ix propagated outwards from a centre in the form of a group, whilst the in- dividual waves composing the group were themselves travelling backwards, coming into existence at the front, and dying out as they approach tho rear *. Moreover, it may be urged that even in the more familiar phenomena of Acoustics and Optics the wave-velocity is of importance chiefly so far as it coincides with the group-velocity. When it is necessary to emphasize the distinction we may borrow the term 'phase-velocity* from modern Physics to denote what is more usually referred to in the present subject as 'wave- velocity.' 238. The theory of the waves produced in deep water by a local dis- turbance of the surface was investigated in two classical memoirs by Cauchy f and Poisson %. The problem was long regarded as difficult, and even obscure, but in its two-dimensional form, at all events, it can be presented in a com- paratively simple aspect. It appears from Arts. 40, 41 that the initial state of the fluid is deter- minate when we know the form of the boundary, and the boundary- values of the normal velocity dcf>/dn, or of the velocity-potential <f>. Hence two forms of the problem naturally present themselves ; we may start with an initial elevation of the free surface, without initial velocity, or we may start with the surface undisturbed (and therefore horizontal) and an initial distribution of surface-impulse (/o</> ). If the origin be in the undisturbed surface, and the axis of y be drawn vertically upwards, the typical solution for the case of initial rest is 7) as cos <rt cos kx, (1) <£ =9 e k v coskx, (2) provided o- 2 = #&, (3) in accordance with the ordinary theory of 'standing waves' of simple- harmonic profile (Art. 228). If we generalize this by Fourier's double-integral theorem f(x) = - dk\ f \a) cos k (x — a) da, (4) then, corresponding to the initial conditions 1 -/(*), 4*> = 0, (5) where the zero suffix indicates surface-value (y = 0), we have 1 f 00 f 00 V = - I cos atdk\ f (a) cos k (x — a) da, (6) "" J J -co" Q^lFE^ekyM r f( a )cosk(x-a)da (7) * Proc. Lond. Math. Soc. (2) i. 473. f I.e. ante p. 17. J "M^moire sur la th^orie des ondes," M€m. de VAcad. Roy. des Sciences, i. (1816). 237-238] Cauchy-Poisson Wave Problem 385 If the initial elevation be confined to the immediate neighbourhood of the origin, so that /(a) vanishes for all but infinitesimal values of a, we have, assuming ° f(a)da = l, (8) — e ky cos kxdk (9) This may be expanded in the form 6 ==^ T \l -~^k + ^k 2 -..\e k y cos kxdk, (10) it Jo I o! 5! J where use is made of (3). If we write — y=r cos 6, x = rsin0, (11) we have, y being negative, e k ^coshxk n dk-- 1 ^ i Qos(n + l)6, (12)* so that (10) becomes at (cos 6 1 /t ~ cos 20 1 71 j9 .„cos30 ) /10X a result which is easily verified. From this the value of rj is obtained by Art. 227 (5), putting = ± |tt. Thus, for x > 0, _ 1 \gt 2 1 (gt*s* 1 (gt*\* \ , ^"tt^^^ "" 3.5V2^/ + 3.5.7.9lW "'} (14JT It is evident at once that any particular phase of the surface disturbance, e.g., a zero or a maximum or a minimum of 77, is associated with a definite value of \gt 2 jx, and therefore that the phase in question travels over the surface with a constant acceleration. The meaning of this somewhat remark- able result will appear presently (Art. 240). The series in (14) is virtually identical with one (usually designated by MX) which occurs in the theory of Fresnel's diffraction-integrals. In its present form it is convenient only when we are dealing with the initial stages of the disturbance; it converges very slowly when \g&\x is no longer small. An alternative form may, however, be obtained as follows. * This formula may be dispensed with. It is sufficient to calculate the value of at points on the vertical axis of symmetry ; its value at other points can then be written down at once by a property of harmonic functions (cf. Thomson and Tait, Art. 498). t That the effect of a concentrated initial elevation of sectional area Q must be of the form V = §f(9t*lx) is evident from consideration of ' dimensions. : J Cf. Kayleigh, Papers, iii. 129. 386 Surface Waves [chap, ix The surface-value of (f> is, by (9), cf>o=-| coakxdk 7tJ o- -j{C^(T + ^) fc "iT™(T" a *)* r } (15) Putti ^ '-?(**©• (16) we find l" sin (— + at) da =^1°° sin (? -co 2 ) d£, (17) [°° sin (--- at\da = 9 - h P° sin (f 2 - a, 2 ) rff , (18) wnere a, = (g)* (19) Hence ^--^P" sin (f 2 - a> 2 ) d£ (20) irx*J o From this the value of 77 is derived by Art. 227 (5); thus 7 7 = ^|r c os(? 2 - G , 2 )^ irx*J = $-\ jcos co 2 P cos (*ae + sin a> 2 P sin J 2 ^} (21) TTX* I JO JO This agrees with a result given by Poisson. The definite integrals are practically of Fresnel's forms*, and may be considered as known functions. Lommel, in his researches on Diffraction f, has given a table of the function ^O + aTTO-"' (22) which is involved in (14), for values of z ranging from to 60. We are thus enabled to delineate the first nine or ten waves with great ease. The figure on the next page shews the variation of 77 with the time, at a particular place ; for different places the intervals between assigned phases vary as six, whilst the corresponding elevations vary inversely as x. The diagrams on p. 388, on * Ju terms of a usual notation we have Pcos t*dt=J(fr) C (u), Psin t*di= V^tt) S (u), Cu fu where C(u)=j cos^-rruPdu, S(w) = | sin ^itu' 2 du, Jo Jo the upper limit of integration being u = ,J(2lir) . w. Tables of C (u) and S (u) computed by Gilbert and others are given in most books on Physical Optics. More extensive tables, due to Lommel, are reproduced by Watson, Theory of Bessel Functions, pp. 744, 745. t "Die Beugungserscheinungen geradlinig begrenzter Schirme," Abh. d. k. Bayer. Akad. d. Wiss. 2° CI. xv. (1886). 238—239] Waves due to a Local Elevation 387 the other hand, shew the wave-profile at a particular instant; at different times, the horizontal distances between corresponding points vary as the square of the time that has elapsed since the beginning of the disturbance, whilst corresponding elevations vary inversely as the square of this time. 1 [The unit of the horizontal scale is JiVxjg). That of the vertical scale is Qlirx, if Q be the sectional area of the initially elevated fluid.] When gt 2 l&x is large, we have recourse to the formula (21), which makes ^-/^-(cosf+sinfY (23) approximately, as found by Poisson and Cauchy. This is in virtue of the known formulae £ cos £*<*?= [J sin ^£=^2 (24) Expressions for the remainder are also given by these writers. Thus Poisson obtains, substantially, the semi-convergent expansion gh < "* 2 2* "Til cos v + sin ~) 4# 4<xJ —'•»©'- This is derived as follows. We have ....(25) .(26) 2ia> ' (2i) 2 co 3 (2*) 3 a> 5 ••*' by a series of partial integrations. Taking the real part, and substituting in the first line of (21), we obtain the formula (25). 239. In the case of initial impulses applied to the surface, supposed undisturbed, the typical solution is p(f> = cos at e ky cos Jcx, (27) 7j = sin cr£ cos for, ,..(28) 99 388 Surface Waves [chap. IX with a 2 = gk as before. Hence, if the initial conditions be p<j>o=F(x), 77 = 0, (29) we have </> = — cosate&dkl F (a) cos k (x - a) da, (30) irpj o J -co rj = <r sin at dk\ F (a) cos k (x — a) da (31) -rrgpJo J -oo ' v } -100- 300- -400 x [The unit of the horizontal scales is \gt\ That of the vertical scales is 2Q/7rgt 2 .] 239] Waves due to a Local Impulse 389 For a concentrated impulse acting at the point x = of the surface, we have, putting 00 F(a)da = l, (32) /: 7rpJo cos ate 1 ® cos kxdk (33) This integral may be treated in the same manner as (9); but it is evident that the results may be obtained immediately by performing the operation 1/gp.d/dt upon those of Art. 238. Thus from (13) and (14) we derive v = COS ^1.2 C08 ^0 irp\~r * 9 r 2 gt M t fi -px 2 (1 irpar (i 1.3.5 \2xJ '1.3 The series in (35) fs related to the function z z z z 5 T7s h~M)'--Y <»>• .(36) 1.3.5.7 1.3.5.7.9.11 which has also been tabulated by Lommel. If we denote the series (22) and (36) by Si and 2 8 , respectively, we find 1 -l^r5 + l.sl7.9 --^ 1+ ^-^ W so that the forms of the first few waves can be traced without difficulty. The annexed figure shews the rise and fall of the surface at a particular 7 60- 50- [The unit of the horizontal scale is J^xjg). That of the vertical scale is P / 2 . / — , where P represents the total initial impulse.] * With the help of the theory of ' dimensions ' it is easily seen a priori that the effect of a concentrated initial impulse P (per unit breadth) is necessarily of the form px 390 Surface Waves [CHAP. IX 0-4 0-5 "-'a? -10000 0-10 AP [The unit of the horizontal scales is \gt 2 . That of the vertical scales is ^ • The upper curve, if continued to the right, would cross the axis of x and would thereafter be indistinguishable from it on the present scale.] 239-240] Interpretation of Results 391 place; for different places the time-intervals between assigned phases vary as \Jx, as in the former case, but the corresponding elevations now vary inversely as x%. In the diagrams on the opposite page, which give an instantaneous view of the wave-profile, the horizontal distances between corresponding points vary as the square of the time, whilst corresponding ordinates vary inversely as the cube of the time. For large values of \gt 2 jx, we find, performing the operation 1/gp.d/dt upon (23), ^- 6 (cosf-sinf), (38) approximately. v = o 240. It remains to examine the meaning and the consequences of the results above obtained. It will be sufficient to consider, chiefly, the case of Art. 238, where an initial elevation is supposed to be concentrated on a line of the surface. At any subsequent time t the surface is occupied by a wave-system whose advanced portions are delineated on p. 388. For sufficiently small values of x the form of the waves is given by (23); hence as we approach the origin the waves are found to diminish continually in length, and to increase continually in height, in both respects without limit. As t increases, the wave-system is stretched out horizontally, proportionally to the square of the time, whilst the vertical ordinates are correspondingly diminished, in such a way that the area Irjdx included between the wave-profile, the axis of x, and the ordinates corre- sponding to any two assigned phases (i.e. two assigned values of a>) is constant*. The latter statement may be verified immediately from the mere form of (14) or (21). The oscillations of level, on the other hand, at any particular place, are represented on p. 387. These follow one another more and more rapidly, with ever increasing amplitude. For sufficiently great values of t, the course of these oscillations is given by (23). In the region where this formula holds, at any assigned epoch, the changes in length and height from wave to wave are very gradual, so that a considerable number of consecutive waves may be represented approxi- * This statement does not apply to the case of an initial impulse. The corresponding pro- position then is that ,dar, taken between assigned values of w, is constant. This appears from (34). ]<Po l 392 Surface Waves [chap, ix mately by a curve of sines. The circumstances are, in fact, all approximately reproduced when *£"** (»») Hence, if we vary t alone, we have, putting At = t, the period of oscillation, T =-^- ; ( 4 °) whilst, if we vary x alone, putting Ax — — X, where \ is the wave-length, we find 8ttx 2 9* The wave-velocity is to be found from X = -3T (41) a£ = 0; (42) ,i • A# 2x /q\ //lox this s ives ^=t = \/L' (43 > by (41), as in the case of an infinitely long train of simple-harmonic waves of length \. We can now see something of a reason why each wave should by con- tinually accelerated. The waves in front are longer than those behind, and are accordingly moving faster. The consequence is that all the waves are continually being drawn out in length, so that their velocities of propagation continually increase as they advance. But the higher the rank of a wave in the sequence, the smaller is its acceleration. So far, we have been considering the progress of individual waves. But, if we fix our attention on a group of waves, characterized as having (approxi- mately) a given wave-length \, the position of this group is regulated according to (43) by the formula ?=*v/fe; ^ i.e. the group advances with a constant velocity equal to half that of the component waves. The group does not, however, maintain a constant amplitude as it proceeds; it is easily seen from (23) that for a given value of A, the amplitude varies inversely as sjx. It appears that the region in the immediate neighbourhood of the origin may be regarded as a kind of source, emitting on each side an endless succession of waves of continually increasing amplitude and frequency, whose subsequent careers are governed by the laws above explained. This persistent activity of the source is not paradoxical; for our assumed initial accumulation of a finite volume of elevated fluid on an infinitely narrow base implies an unlimited store of energy. 24o] Interpretation of Results 393 In any practical case, however, the initial elevation is distributed over a band of finite breadth; we will denote this breadth by I. The disturbance at any point P is made up of parts due to the various elements, 8a, say, of the breadth I; these are to be calculated by the preceding formulae, and integrated over the breadth of the band. In the result, the mathematical infinity and other perplexing peculiarities, which we meet with in the case of a concentrated line-source, disappear. It would be easy to write down the requisite formulae, but, as they are not very tractable, and contain nothing not implied in the preceding statement, they may be passed over. It is more instructive to examine, in a general way, how the previous results will be modified. The initial stages of the disturbance at a distance x, such that Ijx is small, will evidently be much the same as on the former hypothesis; the parts due to the various elements 8a will simply reinforce one another, and the result will be sufficiently expressed by (14) or (23) provided we multiply by / QO f(a) da, -8 i.e. by the sectional area of the initially elevated fluid. The formula (23), in particular, will hold when \gft\x is large, so long as the wave-length \ at the point considered is large compared with I, i.e. by (41), so long as \gt 2 jx . Ijx is small. But when, as t increases, the length of the waves at x becomes comparable with or smaller than I, the contributions from the different parts of I are no longer sensibly in the same phase, and we have something analogous to 'interference' in the optical sense. The result will, of course, depend on the special character of the initial distribution of the values of f{a) over the space I*, but it is plain that the increase of amplitude must at length be arrested, and that ultimately we shall have a gradual dying out of the disturbance. There is one feature generally characteristic of the later stages which must be more particularly adverted to, as it has been the cause of some perplexity; viz. a fluctuation in the amplitude of the waves. This is readily accounted for on 'interference' principles. As a sufficient illustration, let us suppose that the initial elevation is uniform over the breadth I, and that we are considering a stage of the disturbance so late that the value of A, in the neighbourhood of the point x under consideration has become small com- pared with I. We shall evidently have a series of groups of waves separated by bands of comparatively smooth water, the centres of these bands occurring wherever I is an exact multiple of X, say I = ri\. Substituting in (41), we find * 2 V 2mr' ^ 5) i.e. the bands in question move forward with a constant velocity, which is, in * Cf. Burnside, "On Deep-water Waves resulting from a Limited Original Disturbance," Proc. Land. Math. Soc. (1) xx. 22 (1888). 394 Surface Waves [chap, ix fact, the group-velocity corresponding to the average wave-length in the neighbourhood*. The ideal solution of Art. 238 necessarily fails to give any information as to what takes place at the origin itself. To illustrate this point in a special case, we may assume M-it/h* (46) the formula (7) then gives ♦-*/. °° s iE^ e t(»-b) C0S ^^ (47) o a- The surface-elevation at the origin is r, = Q I™ cos <Tte-* b dk=^( C ° cos at £-<*>!* *d<r = ^ % /"" sin ate- **>/<> da. ...(48) ir Jo -ngj o Trgdtjo By a known formula we havet [* > e-* 2 sm2Pxdx=e-f i2 /"" e* 2 dx (49) Hence, putting a> 2 =gt 2 /4:b, (50) we find ri = -% 4- • e" w2 fV*P = X(l~ 2<oe~» 2 [" <* 2 dx\ (51) irb da> Jo tto\ Jo / Hence 4~ (*«**)- -?$■ ["**<&, (52) d(o w ' irb J o shewing that rje^ steadily diminishes as t increases. Hence r] can only change sign once. The form of the integrals in (48) shews that rj tends finally to the limit zero ; and it may be proved that the leading term in its asymptotic value is — 2QJirgt 2 X. One noteworthy feature in the above problems is that the disturbance is propagated instantaneously to all distances from the origin, however great. Analytically, this might be accounted for by the fact that we have to deal with a synthesis of waves of all possible lengths, and that for infinite lengths the wave-velocity is infinite. It has been shewn, however, by Rayleigh § that the instantaneous character is preserved even when the water is of finite depth, in which case there is an upper limit to the wave-velocity. The physical reason of the peculiarity is that the fluid is treated as incompressible, so that changes of pressure are propagated with infinite velocity (cf. Art. 20). When compressibility is taken into account a finite, though it may be very short, interval elapses before the disturbance manifests itself at any point ||. * This fluctuation was first pointed out by Poisson, in the particular case where the initial elevation (or rather depression) has a parabolic outline. The preceding investigations have an interest extending beyond the present subject, as shewing how widely the effects of a single initial impulse in a dispersive medium [i.e. one in which wave-velocity varies with wave-length) may differ from what takes place in the case of sound, or in the vibrations of an elastic solid. The above discussion is taken, with some modifica- tions, from a paper "On Deep- Water Waves," Proc. Lond. Math. Soe. (2) ii. 371 (1904), where also the effect of a local periodic pressure is investigated. t This formula presents itself as a subsidiary result in the process of evaluating I e~ x2 cos 2[3x dx by a contour integration. % The definite integral in (52) has been tabulated by Dawson, Proc. Lond. Math. Soc. (1) xxix. 519 (1898), and the function in (49) by Terazawa, Science Reports of the Univ. of Tokio, vi. 171 (1917). § "On the Instantaneous Propagation of Disturbance in a Dispersive Medium, ...," Phil. Mag. (6), xviii. 1 (1909) [Papers, v. 514]. See also Pidduck, " On the Propagation of a Disturb- ance in a Fluid under Gravity," Proc. Roy. Soc. A, lxxxiii. 347 (1910). || Pidduck, "The Wave-Problem of Cauchy and Poisson for Finite Depth and slightly Com pressible Fluid," Proc. Roy. Soc. A, lxxxvi. 396 (1912). 240-24i] Principle of Stationary Phase 395 241. The space which has been devoted to the above investigation may be justified by its historical interest, and by the consideration that it deals with one of the few problems of the kind which can be solved completely. It was shewn, however, by Kelvin that an approximate representation of the more interesting features can be obtained by a simpler process, which has moreover a very general application*. The method depends on the approximate evaluation of integrals of the type ^ u= I $(m)Jf*dx (1) J a It is assumed that the circular function goes through a large number of periods within the range of integration, whilst (f)(x) changes comparatively slowly; more precisely it is assumed that, when f(x) changes by 2tt, (f>(x) changes by only a small fraction of itself. Under these conditions the various elements of the integral will for the most part cancel by annulling interference, except in the neighbourhood of those values of x, if any, for which f(x) is stationary. If we write x = a + f , where a is a value of x, within the range of integration, such that f (a) = 0, we have, for small values of f , /(*)=/(«) +W(«0, (2) approximately. The important part of the integral, corresponding to values of x in the neighbourhood of a, is therefore equal to A(a)eW P e^w.P^ (3) J -8 approximately, since, on account of the fluctuation of the integrand, the extension of the limits to + 00 causes no appreciable error. Now by a known formula (Art. 238 (24)) we have T e ^*Pd!; = ^ }±r= —-<>*** W J _oo m y/2 m Hence (3) becomes ^ ( a) &m*k* t (5) Vli/"(«)| where the upper or lower sign is to be taken in the exponential according as /" (a) is positive or negative. If a coincides with one of the limits of integration in (1), the limits in (3) will be replaced by and 00 , or — 00 and 0, and the result (5) is to be halved. If the approximation in (2) were continued, the next term would be if 8 /'"( a ); the foregoing method is therefore only valid under the condition * Sir W. Thomson, "On the Waves produced by a Single Impulse in Water of any Depth, or in a Dispersive Medium," Proc. R. S. xlii. 80 (1887) [Papers, iv. 303]. The method of treating integrals of the type (1) had however been suggested by Stokes in his paper ' ' On the Numerical Calculation of a Class of Definite Integrals and Infinite Series," Camb. Tram. ix. (1850) [Papers, ii. 341, footnote]. 396 Surface Waves [chap, ix that %f" (a)//" (a) must be small even when f&f" (a) is a moderate multiple of 2tt. This requires that the quotient /"' («)/{!/" («)l)* should be small. Suppose now that, in a medium of any kind, an initial disturbance, whether of the nature of impulse or displacement, of amount cos kx per unit length, gives rise to an oscillation of the type rj = <j>(k) cos kxe i<rt , (6) where a is a function of h determined by the theory of free waves. The effect of a concentrated unit initial disturbance is then given by the Fourier ex- pression v = ~ [ °° 6 (k) **<**-*■» dk+^- r<b (k) e«»*+**> dk (7) It is understood that in the end only the real part of the expressions is to be retained. The two terms in (7) represent the result of superposing trains of simple- harmonic waves of all possible lengths, travelling in the positive and negative directions of x, respectively. If, taking advantage of the symmetry, we confine our attention to the region lying to the right of the origin, the exponential in the first integral will alone, as a rule*, admit of a stationary value or values, viz. when *£- < 8 > This determines k, and therefore also a, as a function of x and t, and we then find, in accordance with (5), ^ viw%^ r cos( "*~ fa±i,r) ' (9) where the ambiguous sign follows that of d 2 <r/dJ<?. The approximation postu- lates the smallness of the ratio d*a/dk* + *J{t\d*cr/dk 2 \ z } (10) Since .(11) dt by (8), it appears that the wave-length and the period in the neighbourhood of the point x at time t are 27r/k and 27r/cr, respectively. The relation (8) shews that the wave-length is such that the corresponding group-velocity (Art. 236) is xjt. * If the group-velocity were negative, as in some of the artificial cases referred to in Art. 237, the second integral would be the important one. 24l] Graphical Illustrations 397 The above process, and the result, may be illustrated by various graphical constructions*. The simplest, in some respects, is based on a slight modification of the diagram of Art. 236. We construct a curve with X as abscissa and ct as ordinate, where t denotes the time that has elapsed since the beginning of the disturbance. To ascertain the nature of the wave- system in the neighbourhood of any point x, we measure off a length OQ, equal to x, along the axis of ordinates. If PN be the ordinate corresponding to any given abscissa X, the phase of the disturbance at x, due to the elementary wave-train whose wave-length is X, will be given by the gradient of the line QP ; for if we draw QR parallel to ON, we have PR _ PN- OQ = ct-x = *t-kx QR~ ON ~ X ~ 2tt ,(12) Hence the phase will be stationary if QP be a tangent to the curve ; and the predominant wave-lengths at the point x are accordingly given by the abscissae of the points of contact of the several tangents which can be drawn from Q. These are characterized by the property that the group-velocity has a given value x/t. If we imagine the point Q to travel along the straight line on which it lies, we get an indication of the distribution of wave-lengths at the instant t for which the curve has been constructed. If we wish to follow the changes which take place in time at a given point x, we may either imagine the ordinates to be altered in the ratio of the respective times, or we may imagine the point Q to approach in such a way that OQ varies inversely as t. crt=koc * Proc. of the 5th Intern. Congress of Mathematicians, Cambridge, 1912, p. 281. 398 Surface Waves [chap, ix The foregoing construction has the defect that it gives no indication of the relative amplitudes in different parts of the wave-system. For this purpose we may construct the curve which gives the relation between at as ordinate and k as abscissa. If we draw a line through the origin whose gradient is a?, the phase due to a particular elementary wave- train, viz. o-t - kx, will be represented by the difference of the ordinates of the curve and the straight line. This difference will be stationary when the tangent to the curve is parallel to the straight line, i.e. when tdo-\dk — x, as already found. It is further evident that the phase-difference, for elementary trains of slightly different wave-lengths, will vary ultimately as the square of the increment of k. Also that the range of values of k for which the phase is sensibly the same will be greater, and consequently the resulting dis- turbance will be more intense, the greater the vertical chord of curvature of the curve. This explains the occurrence of the quantity td 2 <r/dk 2 in the denominator of the formula (9). In the hydrodynamical problem of Art. 238 we have* *(*) = i, °*=gh (13) whence da\&k = \g±k~\ d 2 <7/dkZ=-lgik-%, d*a/<tt? = $gik-*. ...(14) Hence, from (8), k = gt 2 /4>x 2 , * = gt/2x, (15) and therefore 9** i 77 = ^ pUg<o-t*) V(2?r)«* or, on rejecting the imaginary part, •-&-£-««•) < u > The quotient in (10) is found to be comparable with (2x/gt 2 )%, so that the approximation holds only for times and places such that \gt 2 is large com- pared with x. These results are in agreement with the more complete investigation of Art. 238. The case of Art. 239 can be treated in a similar manner. It appears from (16), or from the above geometrical construction (the curve being now a parabola as in Art. 236), that in the procession of waves at any instant the wave-length diminishes continually from front to rear ; and that the waves which pass any assigned point will have their wave-lengths continually diminishing f. 242. We may next calculate the effect of an arbitrary, but steady, application of pressure to the surface of a stream. We shall consider only the state of steady motion which, under the influence of dissipative forces, * The difficulty as to convergence in this case is met by the remark that the formula (9) of Art. 238 gives 7} = - ^5 = lim y -*. - / e k v cos at cos kx dk, g ot ttJo where y is negative before the limit. t For further applications reference may be made to Havelock, ' « The Propagation of Waves in Dispersive Media...," Proc. Roy. Soc. lxxxi. 398 (1908). 241-242] Surface Disturbance of a Stream 399 however small, will ultimately establish itself*. The question is in the first instance treated directly ; a briefer method of obtaining the principal result is explained in Art. 248. It is to be noted that in the absence of dissipative forces, the problem is to a certain extent indeterminate, for we can always superpose an endless train of free waves of arbitrary amplitude, and of wave-length such that their velocity relative to the water is equal and opposite to that of the stream, in which case they will maintain a fixed position in space. To avoid this indeterminateness, we may avail ourselves of an artifice due to Rayleigh, and assume that the deviation of any particle of the fluid from the state of uniform flow is resisted by a force proportional to the relative velocity. This law of friction does not profess to be altogether a natural one, but it serves to represent in a rough way the effect of small dissipative forces ; and it has the great mathematical convenience that it does not interfere with the irrotational character of the motion. For if we write, in the equations of Art. 6, X = -/jl(u — c), Y= — g — fjLV, Z= — fiw, (1) where c denotes the velocity of the stream in the direction of ^-positive, the method of Art. 33, when applied to a closed circuit, gives jy + fjL\f(uda; + vdy + wdz)=0, (2) whence j{udx + vdy + wdz) = Ge'^ (3) Hence the circulation in a circuit moving with the fluid, if once zero, is always zero. We now have - = const. - gy + /jl (ex + $) — J<? a , (4) this being, in fact, the form assumed by Art. 21 (2) when we write 0, = gy-fi(cx-h(f>) (5) in accordance with (1) above. To calculate, in the first place, the effect of a simple-harmonic distribution of pressure we assume - = - x + &e k y sin Jcx, J^ = — y + ft e k y coskx (6) * The first steps of the following investigation are adapted from a paper by Rayleigh, "The Form of Standing Waves on the Surface of Eunning Water," Proc. Lond. Math. Soc. xv. 69 (1883) [Papers, ii. 258], being simplified by the omission, for the present, of all reference to Capillarity. The definite integrals involved are treated, however, in a somewhat more general manner, and the discussion of the results necessarily follows a different course. The problem had been treated by Popoff, "Solution d'un probleme sur les ondes permanentes," Liouville (2), iii. 251 (1858); his analysis is correct, but regard is not had to the indeterminate character of the problem (in the absence of friction), and the results are consequently not pushed to a practical interpretation. 400 Surface Waves [chap, ix The equation (4) becomes, on neglecting the square of &/3, ■£- = . . . — gy + fie ky (kc 2 cos kx -f fie sin kx) (7) This gives for the variable part of the pressure at the upper surface (^r = 0) p = p/3 {(kc 2 — g) cos kx + yuc sin kx), (8) which is equal to the real part of p8 (kc 2 - g - ific) e ikx . If we equate the coefficient of e ikx to C, we may say that to the pressure po^Ce*** (9) corresponds the surface-form w-sr^ft*. (io) where we have written k for g/c 2 , so that Iitjk is the wave-length of the free waves which could maintain their position against the flow of the stream. We have also put A t / C== / U i> f° r shortness. Hence, taking the real parts, we find that the surface-pressure p = Ccoskx (11) produces the wave -form ~ (k — k) cos kx — /«<! sin kx „ _ N m-*o* — w=WTk (12) This shews that if jjl be small the wave-crests will coincide in position with the maxima, and the troughs with the minima, of the applied pressure, when the wave-length is less than 2ir//c; whilst the reverse holds in the opposite case. This is in accordance with a general principle. If we impress on everything a velocity — c parallel to x, the result obtained by putting /i! = in (12) is seen to be a special case of Art. 168 (14). In the critical case of k = k, we have gpy = -—. sinkx, (13) Pi shewing that the excess of pressure is now on the slopes which face down the stream. This explains roughly how a system of progressive waves may be maintained against our assumed dissipative forces by a properly adjusted distribution of pressure over their slopes. 243. The solution expressed by (12) may be generalized, in the first place by the addition of an arbitrary constant to x, and secondly by a sum- mation with respect to k. In this way we may construct the effect of any arbitrary distribution of pressure, say Po=A°>) (14) with the help of Fourier's Theorem (Art. 238 (4)). 242-243] Surface Distribution of a Stream 401 We will suppose, in the first instance, that f(os) vanishes for all but infinitely small values of a?, for which it becomes infinite in such a way that P f{x)dx = P\ (15) J -oo' this will give us the effect of an integral pressure P concentrated on an infinitely narrow band of the surface at the origin. Replacing C in (12) by Pjir . &k, and integrating with respect to k between the limits and oo , we obtain kP f°° (k — tc)coskx — /xisin&# 77 /nn -. goy = — . j~i * 2 dk (16) »™ 7T Jo {k-Kf + fX^ If we put £=k + im, where k, m are taken to be the rectangular co-ordinates of a variable point in a plane, the properties of the expression (16) are contained in those of the complex integral / e ix C f=i* < 17 > It is known that the value of this integral, taken round the boundary of any area which does not include the singular point (f=c), is zero. In the present case we have c = k + ifii , where < and m are both positive. Let us first suppose that x is positive, and let us apply the above theorem to the region which is bounded externally by the line m = and by an infinite semicircle, described with the origin as centre on the side of this line for which m is positive, and internally by a small circle surrounding the point (*, m). The part of the integral due to the infinite semicircle obviously vanishes, and it is easily seen, putting £ — c = re id , that the part due to the small circle is if the direction of integration be chosen in accordance with the rule of Art. 32. We thus obtain [0 fiikx /"oo gikx i — ; ^- N dh + I -j — -. — . dk-27rie i ( K + i ^) x =0, J -oDJC-iK+lpi) JO k-(K+lm) which is equivalent to /"oo pikx /"oo fi-ikx Jo k-(K+im) Jo A: + (k + i/xi) On the other hand, when x is negative we may take the integral (17) round the contour made up of the line ra=0 and an infinite semicircle lying on the side for which m is negative. This gives the same result as before, with the omission of the term due to the singular point, which is now external to the contour. Thus, for x negative, /"oo pikx fee p -ikx r-j- — v—.dk=\ y-i —dk (19) Jo £-(«c+l/ii) JoH(k+^/Xi) V An alternative form of the last term in (18) may be obtained by integrating round the contour made up of the negative portion of the axis of k, and the positive portion of the axis of m, together with an infinite quadrant. We thus find /"O qVcx Too p-mx I i — i — v~- — sdk+ \ / . . idm=0, J -azk-(K+iiLi) Jo zm-(/c + i/x 1 ) which is equivalent to Too e - ilex /"oo Jo £ + (k-M>i) ' ~ J fll+U dm (20) 402 Surface Waves [chap, ix This is for x positive. In the case of x negative, we must take as our contour the negative portions of the axes of k, ra, and an infinite quadrant. This leads to fco p — ikx fco pvnx j-^ r-,dk= — ~ rdm, (21) as the transformation of the second member of (19). In the foregoing argument /^ is positive. The corresponding results for the integral pixg , r . s dc (22) are not required for our immediate purpose, but it will be convenient to state them for future reference. For x positive, we find ,ikx fco p — ikx fco p-mx dm; (23) / fco oikx fco Q — ikx Too jofc-(K-im) ~Jo £+(k-«>i) ~Jo m+ni + iK whilst, for x negative, oikx fco p-ikx fco oikx fco Q-xkx I -. — -. ;— r dk = - 2irie* (« - <Mi) * + j - — — dk jo*-(k-»/*i) Jok + (K-ifj fco omx = - 2rrie i (*- i i*i) x + \ dm (24) Jo m — pi — iK The verification is left to the reader*. If we take the real parts of the formulae (18), (20), and (19), (21), respectively, we obtain the results which follow. The formula (16) is equivalent, for x positive, to trap rt „ . f°° (k + k) cos kx — iii sin kx 77 kP * Jo (& + «) +A*1 o ,. *• • , f 00 (m - vi) e~ mx dm /oex = -2ire-WsmKX+ ±- ™ , (25) Jo (m-fi^f+tc 2 and, for x negative, to 1T9P „_ l m (m-h H )e mx dm . KP' y -J (m + rip + * K0) The interpretation of these results is simple. The first term of (25) represents a train of simple-harmonic waves, on the down-stream side of the origin, of wave-length 2irc?lg, with amplitudes gradually diminishing according to the law e~^ x . The remaining part of the deformation of the free surface, expressed by the definite integrals in (25) and (26), though very great for small values of x, diminishes very rapidly as x increases in absolute value, however small the value of the- frictional coefficient fi lt When ixx is infinitesimal, our results take the simpler forms Trap _ . [ m coskx dk f °° Ytl6~ mx — — 2tt sin kx 4- — % ^dm t (27) JO Wl T * * For another treatment of these integrals, see Dirichlet, Vorlesungen ueber d. Lehre v. d. einfachen u. mehrfachen bestimmten Integralen (ed. Arendt), Braunschweig, 1904, p. 170. 243-244] Surface Disturbance of a Stream 403 for x positive, and cos Am? <irgp f 00 cos for 77 f°° me mx , ,_ ox kP u Jo k + /c Jo m a + « 2 v 7 for x negative. The part of the disturbance of level which is represented by the definite integrals in these expressions is now symmetrical with respect to the origin, and diminishes constantly as the distance from the origin increases. When kx is moderately large we find, by usual methods, the semi-convergent expansion (•00 me -mx l 3 J 5 j ^ s =-srr 2 --4i4+ -ins- ( 29 ) Jo m 2 +K 2 tc 2 x 2 k*x* k q x 6 It appears bhat at a distance of about half a wave-length from the origin, on the down-stream side, the simple-harmonic wave-profile is fully established. The definite integrals in (27) and (28) can be reduced to known functions as follows. If we put (k + k) x=u, we have, for x positive, /°° cos kx _, _ Z" 00 cos {kx — u), o k+K ~ J KX u = -CiKXCOHicx + (%ir-Si <x)smKX, (30) where, in conformity with the usual notation, Ciw=-| du, Siu=l du (31) Ju u Jo u The functions Ci u and Si u have been tabulated by Glaisher*. It appears that as u increases from zero they tend very rapidly to their asymptotic values and £77, respectively. For small values of u we have Ci u=y + log u — 2.2! " 4.4! SlU=*U — V? u .(32) 3.3! ' 5.5! ' where y is Euler's constant *5772.... It is easily found from (25) and (26) that when fi\ is infinitesimal, the integral depression of the surface is -[" **»-£, (33) j -00 yp exactly as if the fluid were at rest. 244. The expressions (25), (26) and (27), (28) alike make the elevation infinite at the origin, but this difficulty disappears when the pressure, which we have supposed concentrated on a mathematical line of the surface, is diffused over a band of finite breadth. * "Tables of the Numerical Values of the Sine-Integral, Cosine-Integral, and Exponential Integral," Phil. Trans. 1870; abridgments are given by Dale and by Jahnke and Emde. The expression of the last integral in (27) in terms of the sine- and cosine-integrals was obtained, in a different manner from the above, by Schlomilch, " Sur 1' integrate d^finie / ^ ^e~^, J d 2 + a? xxxiii. (1846); see also De Morgan, Differential and Integral Calculus, London, 1842, p. 654, and Dirichlet, Vorlesungen, p. 208. 404 Surface Waves [chap, ix To calculate the effect of a distributed pressure J>o =/(*), (34) it is only necessary to write x — a for x in (27) and (28), to replace P by f(a) 8a, and to integrate the resulting value of y with respect to a between the proper limits. It follows from known principles of the Integral Calculus that if p be finite the integrals will be finite for all values of a?. In the case of a uniform pressure p , applied to the part of the surface extending from — oc to the origin, we easily find by integration of (25), for x> 0, . *p f 00 e- mx dm gpy = -2p cos K x + ^j^ m * + H a > ( 3o ) where yui has been put = 0. Again, if the pressure p be applied to the part of the surface extending from to + oo , we find, for x < 0, /cpo f °° em x dm f°° em x dm /Q ~, J. isrn? (36) From these results we can easily deduce the requisite formulae for the case of a uniform pressure acting on a band of finite breadth. The definite integral in (35) and (36) can be evaluated in terms of the functions Ci u, Si u ; thus in (35) [™e- mx dm r 00 sin foe,, rl a . N , n . . . /Qh .. /c — s 5 = i dk = (*7r — Si ##) cos /ca? + Ci /e# sin /c^c. . . .(37) Jo m 2 + « 2 Jo H« In this way the diagram on p. 405 was constructed ; it represents the case where the band (AB) has a breadth k~\ or 159 of the length of a standing wave. The circumstances in any such case might be realized approximately by dipping the edge of a slightly inclined board into the surface of a stream, except that the pressure on the wetted area of the board would not be uniform, but would diminish from the central parts towards the edges. To secure a uniform pressure, the board would have to be curved towards the edges, to the shape of the portion of the wave-profile included between the points A, B in the figure. It will be noticed that if the breadth of the band be an exact multiple of the wave-length (27r//c), we have zero elevation of the surface at a distance, on the down-stream as well as on the up-stream side of the source of disturbance. The diagram shews certain peculiarities at the points A, B due to the discontinuity in the applied pressure. A more natural representation of a local pressure is obtained if we assume p>=ivr* < 38 > 244] Wave-Profile 405 406 Surface Waves [chap, ix We may write this in the form PI PC™ p = i-._!^ = £ e~ kb+ikx dk, (39) r 7T 6 — IX 7tJ provided it is understood that, in the end, only the real part -is to be retained. On reference to Art. 242 (9), (10), we see that the corresponding elevation of the free surface is given by „ p r<x> p— kb+ikx gpy = —\ .— r-dk (40) By the method of Art. 243, we find that this is equivalent, for x > 0, to „.p ( foo p—imb—mx gp y = ?£-\ 2irie^ +i ^ <**"*> 4- — dm\ , (41) and, for a? < 0, to 0py = — — ■ —dm (42) Hence, taking real parts, and putting ^ = 0, we find r. r» * • «■? f °° w* cos mb — k sin mb m „ ■. r ~ n grpy = - 2fcPe- Kh sin k# H s s e~ mx dm, [x > 01, yrj/ 7r J o m 2 + tc 2 L J (43) /cP f 00 m cos mb — K sin m& „,„ 7 r „-. w=~J o ^^ *"*». [*«>]. (44) The factor e~ Kb in the first term of (43) shews the effect of diffusing the pressure. It is easily proved that the values of y and dy/dx given by these formulae agree when x—0*. 245. If in the problem of Art. 242 we suppose the depth to be finite and equal to h, there will be, in the absence of dissipation, indeterminateness or not, according as the velocity c of the stream is less or greater than (gh)%, the maximum wave-velocity for the given depth; see Art. 229. The difficulty presented by the former case can be evaded by the introduction of small frictional forces; but it m&y be anticipated from the preceding investigation that the main effect of these will be to annul the elevation of the surface at a distance on the up-stream side of the region of disturbed pressure, and if we assume this at the outset we need not complicate our equations by retaining the frictional terms f. For the case of a simple-harmonic distribution of pressure we assume 0. .(1) ■#+j3 cosh k (y + h) sin hx, — = — ;/+/3 sinh k (y + h) cos kx, * A different treatment of the problem of Arts. 243, 244 is given in a paper by Kelvin, "Deep Water Ship-Waves," Proc. R. S. Edin. xxv. 562 (1905) [Papers, iv. 368]. t There is no difficulty in so modifying the investigation as to take the frictional forces into account, when these are very small. 244-245] Stream of Finite Depth 407 as in Art 233 (3). Hence, at the surface y=£ sinh M cos &r, (2) we have — = - gy - \ (q 2 - c 2 ) = 8 {Tec 2 cosh kh-gsinhkh) cos kx, (3) p so that to the imposed pressure p =Ccoskx (4) will correspond the surface-form __C sinh kh , ,_, p ' kc 2 cosh kh—g sinh kh ' As in Art. 242, the pressure is greatest over the troughs, and least over the crests, of the waves, or vice versd, according as the wave-length is greater or less than that corresponding to the velocity c, in accordance with general theory. The generalization of (5) by Fourier's method gives _ P /"°° sinh kh cos kx „ , ft , * irp J o kc 2 cosh kh - g sinh kh as the representation of the effect of a pressure of integral amount P applied to a narrow band of the surface at the origin. This may be written ttoc 2 [*> cos (xu/h) , mx -r ■ y= }« uoothu-ghi<? da (7) Now consider the complex integral 9 ixClh dt, (8) /; ^cothC-; where £=u + iv. The function under the integral sign has a singular point at (=+ico, according as x is positive or negative, and the remaining singular points are given by the roots of tanh£_ c 2 ~1 gk (9) Since (6) is an even function of x, it will be sufficient to take the case of x positive. Let us first suppose that c 2 > gh. The roots of (9) are then all pure imaginaries ; viz. they are of the form ± ij3, where j3 is a root of ¥*£ ™ The smallest positive root of this lies between and ^tt, and the higher roots approximate with increasing closeness to the values (s + ^) it, where s is integral. We will denote these roots in order by /3 , j3j, j8 2 , .... Let us now take the integral (8) round the contour made up of the axis of u, an infinite semicircle on the positive side of this axis, and a series of small circles surrounding the singular points {=ip , ifii, $3 2 , The part due to the infinite semicircle obviously vanishes. Again, it is known that if a be a simple foot of f(£) = the value of the integral / /«>* taken in the positive direction round a small circle enclosing the point £=a is equal to -f§ <») Now in the case of (8) we have /'(a) = cotha-a(coth2 a -l)=i|^(l-^) + a'j, (12) whence, putting a=i/3 8 , the expression (11) takes the form 27rB 8 e-f*s*/\ (13) /; 408 Surface Waves [chap, ix where *. " wn w\ (14) P* " c 2 \ C 2 ) The theorem in question then gives f0 fiixu'h /"co e ixu/h ^ J -ooWcothw-^A/c 2 Joucothu-gh/c 2 o If in the former integral we write — m for m, this becomes cos (xulh) 7 "C 00 D a v/ fc /t/>\ — ^ — L_^ ^ = 7r 2 n B s e-Ps x ' h (16) o « coth u— ghj c 2 ° The surface-form is then given by y=j?-K B > e ~ f ' ,xlh (17) It appears that the surface-elevation (which is symmetrical with respect to the origin) is insensible beyond a certain distance from the seat of disturbance. When, on the other hand, c 2 <gh, the equation (9) has a pair of real roots ( + a, say), the lowest roots (±/3 ) of (10) having now disappeared. The integral (7) is then indeterminate, owing to the function under the integral sign becoming infinite within the range of integration. One of its values, viz. the ' principal value,' in Cauchy's sense, can however be found by the same method as before, provided we exclude the points £= ±o from the contour by drawing semicircles of small radius e round them, on the side for which v is positive. The parts of the complex integral (8) due to these semicircles will be g±iax/h where/' (a) is given by (12); and their sum is therefore equal to 27r^lsiny, (18) where A= (19) The equation corresponding to (16) now takes the form ( fa-e [co ] cos l X u\h) , . . (XX ^oo B . . . { + Y ttt — l -~T- 2 du = -7rAsm- r + 7rX B g e~^ h , (20) (Jo J a+e) u coth u-gh/c 2 hi 8 ' v ' so that, if we take the principal value of the integral in (7), the surface-form on the side of x positive is y=_^ 8in T + 42 S r^«" P '** (21) pc* h pc* • . l Hence at a distance from the origin the deformation of the surface consists of the simple-harmonic train of waves indicated by the first term, the wave-length 2ttA/o being that corresponding to a velocity of propagation c relative to still water. Since the function (7) is symmetrical with respect to the origin, the corresponding result for negative values of x is '-p^T + J 2 !"*** < 22 > The general solution of our indeterminate problem is completed by adding to (21) and (22) terms of the form Ccos™ + Dsin™ (23) 245-246] Irregularities in the Bed of a Stream 409 The practical solution, including the effect of infinitely small dissipative forces, is obtained by so adjusting these terms as to make the deformation of the surface insensible at a distance on the up-stream side. We thus get, finally, for positive values of x, y--^Bh? + — -S"*.*-^. (24) 9 pc 2 h pc 2 l and, for negative values of x, y-^2^.A-» (25) For a different method of reducing the definite integral in this problem we must refer to the paper by Kelvin cited below. 246. The same method can be employed to investigate the effect on a uniform stream of slight inequalities in the bed*. Thus, in the case of a simple-harmonic corrugation given by y = —h + y cos kx, (1) the origin being as usual in the undisturbed surface, we assume — = - x + (a cosh ky + {$ sinh ky) sin kx, -.(2) — = - y + (a sinh ky + (3 cosh ky) cos kx. The condition that (1) should be a stream-line is y= - a sinh kh + /3 cosh kh (3) The pressure-fonnula is - = const. -gy + kc 2 (o cosh ky + /3 sin ky) cos kx, (4) approximately, and therefore along the stream-line yjs = - = const. 4- (kc 2 a —gfi) cos kx, so that the condition for a free surface gives kc 2 a -gp = Q (5) The equations (3) and (5) determine a and 0. The profile of the free surface is given by '■ ffw< " a]FfeiB w '*' (6) If the velocity of the stream be less than that of waves in still water of uniform depth h, of the same length as the corrugations, as determined by Art. 229 (4), the denominator is negative, so that the undulations of the free surface are inverted relatively to those of the bed. In the opposite case, the undulations of the surface follow those of the bed, but with a different vertical scale. When c has precisely the value given by Art. 229 (4), the solution fails, as we should expect, through the vanishing of the denominator. To obtain an intelligible result in this case we should be compelled to take special account of dissipative forces. The above solution may be generalized, by Fourier's Theorem, so as to apply to the case where the inequalities of the bed follow any arbitrary law. Thus, if the profile of the bed be given by y=-h+f(x)=-h+~f a> dk[ CO /(£) cos A (*-£)<*£, (7) 7Tj J -00 * Sir W. Thomson, "On Stationary Waves in Flowing Water," Phil. Mag. (5) xxii. 353, 445, 517 (1886), and xxiii. 52 (1887) [Papers, iv. 270]. The effect of an abrupt change of level in the bed is discussed by Wien, Hydrodynamik, p. 201. 410 Surface Waves [chap, ix that of the free surface will be obtained by superposition of terms of the type (6) due to the various elements of the Fourier-integral; thus If 00 f 00 /(flooB*(*-e dt oo cosh kh — g/kc 2 . sinh kh In the case of a single isolated inequality at the point of the bed vertically beneath the origin, this reduces to _ Q /• COS&37 „ 7T J o cosh kh — g/kc 2 . sinh kh irh) o ucos(xu/h) j_ m w cosh u — gh/c 2 . sinh u where Q represents the area included by the profile of the inequality above the general level of the bed. For a depression Q will of course be negative. The discussion of the integral I fi M *"*f (10) J C cosh C-ghjc 2 . sinh £ v } can be conducted exactly as in Art. 245. The function to be integrated differs only by the factor £/(sinh £) ; the singular points therefore are the same as before, and we can at once write down the results. Thus when c 2 > gh we find, for the surface- form, y=$2*B 8 -^- e*M\ (11) 9 h 8 sin ft, ' v ' the upper or the lower sign being taken according as x is positive or negative. When c 2 <gh, the 'practical' solution is, for x positive, y- fA*M% + f2"B*e-**l> 9 (12) A sinh a h h i sin/3 8 and, for x negative, y=-^'£ co B 8 -J-?- e^ x ' h (13) rt l sin pig The symbols a, /3 S , A, B 8 have here exactly the same meanings as in Art. 245 * 247. We may calculate, in a somewhat similar manner, the disturbance produced in the flow of a uniform stream by a submerged cylindrical obstacle whose radius b is small compared with the depth/ of its axisf. The cylinder is supposed placed horizontally athwart the stream. We write / b\ 4>~-cx\l +;i)+J6 (!) where c denotes as before the general velocity of the stream, and r denotes distance from the axis of the cylinder, viz. r = V(^ + (y+/) 2 ), (2) * A very interesting drawing of the wave-profile produced by an isolated inequality in the bed is given in Kelvin's paper, Phil. Mag. (5) xxii. 517 (1886) [Papers, iv. 295]. + The investigation is taken from a paper "On some cases of Wave-Motion on Deep Water," Ann. di matematica (3), xxi. 237 (1913). I find that the problem had been suggested by Kelvin, Phil. Mag. (6) ix. 733 (1905) [Papers, iv. 369]. 246-247] Waves due to a Submerged Cylinder 411 the origin being in the undisturbed level of the surface, vertically above the axis. This makes d<f)/dr = for r = b, provided x De negligible in the neigh- bourhood of the cylinder. We assume X=\ a(k)e^y sin kccdk, (3) Jo where a (k) is a function of k, to be determined. For the equation of the free surface, assumed to be steady, we put V = 8 (k) cos kxdk (4) Jo The geometrical condition to be satisfied at the free surface is -£-?£. < 5 > wherein we may put y = 0. Since (1) is equivalent to r°° (}) = -cx-b 2 c e- k( v+f> sin kacdk+x> ( 6 ) Jo for positive values of y +/, this condition is satisfied if b 2 ce-W + a(k) = cj3(k) (7) Again, the variable part of the pressure at the free surface is given by f 00 dy — — 9V — i° 2 — b 2 c 2 e~ kf cos kxkdk + c =* = — 9V "" i ° 2 ~ & 2c2 e _A;/ cos kxkdk + c a (&) cos kxkdk, (8) Jo Jo where terms of the second order in the disturbance have been omitted. This expression will be independent of x provided g/3(k) + kb 2 c 2 e- k f-kca(k) = (9) Combined with (7), this gives «(*)-£^P<*-V ^W = fr"^ » ( 10 ) where K = g/c 2 , (11) as in Art. 242. Hence ^70 f°° ke~ k f cos kxdk 77 = 26 2 ^— - --d>^<^^ w 412 Surface Waves [chap, ix The integral is indeterminate, but if x be positive its principal value is equal to the real part of the expression iire- K ^ iKX + i\ -. dm (13) Jo im—/c Adopting this we have 2b 2 f 7) — 2 ,/ 2 — 2ir K,b 2 e~ Kf sin kx - 2*6* r (K Si ° mf ~ T— z mf) ^ dm (14) Jo m 2 + /c 2 For large values of x the second term is alone sensible. Since the value of tj in (12) is an even function of x we must have, for x negative, 2 &Y , o z,2 - K f - o z.2 f°°(« sin rnf-m cos mf)e mx , v ^= 2 ^ + %Tr/cb 2 e v sm k x- 2 Kb 2 \ v * — ^ , — — — am. ...(15) x 2 +f 2 Jo ra 2 +/e 2 On the disturbances represented by these formulae we can superpose any system of stationary waves of length 2ir/fc, since these could maintain their position in space, in spite of the motion of the stream ; and if we choose as our additional system 7] = — 2tt Kb 2 e~ K f sin kx (16) we shall annul the disturbance at a distance on the up-stream side (x < 0), as is required for a physical solution. The result is 2b 2 f 7] = -g — ~ - ^iTKb 2 e~ Kf sin kx -f &c. [x > 0], It appears that there is a local disturbance immediately above the obstacle, followed by a train of w$ves of length 2irc 2 lg on the down-stream side *. The investigation is easily adapted to the case where the section of the cylinder has any arbitrary form. The assumption really made above is that, to a first approximation, the effect of the cylinder at a distance is that of a suitably adjusted double source. In the more general case, referring to Art. 72 a, we may write 4>= -ftp + ^ + x, (18) *--<%&$&&• < 19) It is convenient to work with complex quantities, and to write .(17) 1 =cf C °e- l (i/+/) + «*rfi (20) with g= ^ + <?)-g (21) 2tt * If we investigate the asymptotic expansion of the definite integral in (13), when k/ is large, we find on substitution in (12) that the most important term gives - 2b 2 //(:r 2 +/ 2 ), and so cancels the first term in the above values of r). The approximation has been carried further, for moderate values of k/, by Havelock, Proc. Roy. Soc. A, cxv. 274 (1927). 247-248] Effect of a Travelling Disturbance 413 The real part of (20) is of course alone to be retained in the end. The steps of the calcu- lation may be supplied by the reader. The final result is, for large values of | x |, ■ ( A+ .f/ ^ X -{2(A + Q)KsmKX + 2KHcosKx}e-*f+&c. [a?>0], (22) The local disturbance near the origin is not symmetrical unless H=0. For an elliptic section whose major axis makes an a.ngle a with the direction of the stream, we have ^l = 7 r(a 2 sin* 2 a + 6 2 cos 2 a), Q = nab, H= it (a 2 - 6 2 ) sin a cos a (23) The square of the amplitude of the waves is then 4/c 2 (^ + ^) 2 + 4/c 2 .e' 2 = 47r 2 K 2 (a + 6) 2 (a 2 sin 2 a + 6 2 cos 2 a) (24) 248. If in the problems of Arts. 243, 245 we impress on everything a velocity — c parallel to x, we get the case of a pressure-disturbance advancing with constant velocity c over the surface of otherwise still water. In this form of the question it is not difficult to understand, in a general way, the origin of the train of waves following the disturbance. If, for example, equal infinitesimal impulses be applied in succession to a series of infinitely close equidistant parallel lines of the surface, at equal intervals of time, each impulse will produce on its own account a system of waves of the character investigated in Art. 239. The systems due to the different impulses will be superposed, with the result that the only parts which reinforce one another will be those whose wave-velocity is equal to the velocity c with which the disturbing influence advances over the surface, and which are (moreover) travelling in the direction of this advance. And the investigations of Arts. 236, 237 shew that in the present problem the groups of waves of this particular length which are produced are continually being left behind. When capillary waves come to be considered, the latter statement will need to be modified. The question can be investigated from a general standpoint, independent of the particular kind of waves considered, as follows *. We take the origin at the instantaneous position of the disturbing influence, which is supposed to travel with velocity c in the direction of ^-negative. The effect of an impulse Bt delivered at an antecedent time t is given by Art. 241 (7) if we replace x by ct — x and multiply by St. Introducing $he hypothesis of a small frictional force varying as the velocity, and integrating from t = to t—QC, we get v =±- l w | r $ (k) e i7t ~ ik < c< -*> dk + j °°0 (k)e iTt+ik < c '-*> dk\ e^* dt. . . .(1) integration with respect to t gives _ 1 f 00 cj>(k)e ikx dk 1 p (f>(k)e~ ikx dk 2irJo \ii — i{(T — kc) 27rJ \i*< — i(<r + kc) ' * Phil. Mag. (6) xxxi. 386 (1916). 414 Surface Waves [chap, ix The quantity //- is by hypothesis small, and will in the limit be made to vanish. The most important part of the result will therefore be due to values of k in the first integral which make 5 * = kc (3) approximately. Writing k = /c + k', where k is a root of this equation, we have a-kc = (^-cy , = {U-c)k f , (4) nearly, where U denotes the group- velocity corresponding to the wave-length 2tt/k. The important part of (2) for large values of x is therefore 1 f oo pik'x /JL' '-ir + W-L b-W-OV < 5 > since the extension of the range of integration to k' — ± oo makes no serious difference. Now if a be positive we have* roo e imx^ m ^ Mire-**, [x>0] J_ooa-Mm ( 0, |><0] {) whiist j^^fa"ishrt- [«<o] (7) Hence if U<c W)*™ e-triie-U) or o, (8) c— U according as x ^ ; whilst if U> c v = 0, or ^^e-W(^) (9) in the respective cases. If we now make yu. -^ we have the simple expression ^ = -|7^Ff (10) for the wave-train generated by the travelling disturbance. This train follows or precedes the disturbing agent acccording as U $ c. Examples of the two cases are furnished by gravity waves on water, and capillary waves, respectively (Arts. 236, 266). The approximation in (4) is valid only if the quotient d*a/dk\k'~r-(U-c) (11) is small even when k'x is a moderate multiple of 27r. This requires that d*cr/dk* + (U-c)x (12) should be small. Unless JJ—c, exactly, the condition is always fulfilled if x be sufficiently great. It may be added that the results (8), (9) are accurate, in the sense that they give the leading term in the evaluation of (2) by Cauchy's method of residues. Cf. Art. 242. * The results quoted are equivalent to the familiar formulae /" cos mx dm _ f" m sin mx dm a^ 1 tf-Htf =" (where the upper or lower sign is to be chosen according as x is positive or negative), but can be obtained directly by contour integration. 248-249] Wave-Resistance 415 In the case of waves on deep water, due to a concentrated pressure of integral amount P, we put <f>(k) = icrP/gp, (13) to conform to Art. 239 (28). Since U = Jc, we obtain, on taking the real part, ?7 = sin kx, (14) 9P in agreement with (27) of Art. 243*. If there is more than one value of k satisfying (3), there will be a term of the type (10) for each such value. This happens in the case of water-waves due to gravity and capillarity combined (Art. 269), and in the case of super- posed fluids, to be referred to presently. 249. The preceding results have a bearing on the theory of ' wave-resistance.' Taking the two-dimensional form of the question, let us imagine two fixed vertical planes to be drawn, one in front, and the other in the rear, of the disturbing body. If U < c the region between the planes gains energy at the rate cE, where E is the mean energy per unit area of the free surface. This is due partly to the work done at the rear plane, at the rate UE (Art. 237), and partly to the reaction of the disturbing body. Hence if R be the resistance experienced by the latter, so far as it is due to the formation of waves, we have Rc+UE = cE, or R=°^E. (1) c On the other hand, if U > c, so that the wave-train precedes the body, the space between the planes loses energy at the rate cE. Since the loss at the first plane is UE, we have Rc-UE=-cE, or R=^^E. (2) c Thus, in the case of a disturbance advancing with velocity c [< y/(gh)] over still water of depth h, we find, on reference to Art. 237, R -M 1 -!™)' (3) where a is the amplitude of the waves. As c increases from to y/(gh), kK diminishes from oo to 0, so that R diminishes from %gpa 2 to 0. When c > *J(gh), the effect is merely local, and R = f. It must be remarked, however, that the amplitude a due to a disturbance of given type will also vary with c. For instance, in the case of the submerged cylinder, Art. 244 (43), a varies as ice~ Kh , where k = gjc 2 , the depth being infinite. Hence R varies as c -4g-2^/c2 ( 4 )J * It is not difficult to derive from (2) the complete formula referred to. f Cf. Sir W. Thomson, "On Ship Waves," Proc. Inst. Mech. Eng. Aug. 3, 1887 [Popular Lectures and Addresses, London, 1889-94, iii. 450]. A formula equivalent to (3) was given in a paper by the same author, Phil. Mag. (5) xxii. 451 [Papers, iv. 279]. X The vertical force on the cylinder is calculated by Havelock, Proc. Roy. Soc. A, cxxii. 387 (1928). 416 Surface Waves [chap. IX An interesting variation of the general question is presented when we have a layer of one fluid on the top of another of somewhat greater density. If p, p' be the densities of the lower and upper fluids, respectively, and if the depth of the upper layer be A', whilst that of the lower fluid is practically infinite, the results of Stokes quoted in Art. 231 shew that two wave-systems may be generated, whose lengths (27r//«) are related to the velocity c of the disturbance by the formulae C >=Z, <*= P-P' , i (5, K pCOth K/l+p K It is easily proved that the value of < determined by the second equation is real only if c*< p-p .(6) If c exceeds the critical value thus indicated, only one type of waves will be generated, and if the difference of densities be slight the resistance will be practically the same as in the case of a single fluid. But if c fall below the critical value, a second type of waves may be produced, in which the amplitude at the common boundary greatly exceeds that at the upper surface ; and it is to these waves that the ' dead-water resistance ' referred to in Art. 231 is attributed*. The problem of the submerged cylinder (Art. 247) furnishes an instance where the wave-resistance to the motion of a solid can be calculated. The mean energy, per unit area of the water surface, of the waves represented by the second term in equation (14) of that Art. is E=\gp(£TTK&e-*f)\ Since U=\c, we have from (1) R^AntgpbWe-W. (7) For a given depth (/) of immersion, this is greatest when <f— 1, or , , , . t c-J(gf) (8) In terms of the velocity c we have R=47rYpbK c-te-W* (9) The graph of R as a function of c is appended f. R *(gt) * Ekman, I.e. ante p. 371. t Ann. di mat.. I.e. See also the paper by the author, there quoted. 249-250] Wave-Resistance 417 Waves of Finite Amplitude. 250. The restriction to ' infinitely small ' motions, in the investigations of Arts. 227, ..., implies that the ratio (ajX) of the maximum elevation to the wave-length must be small. The determination of the wave-forms which satisfy the conditions of uniform propagation without change of type, when this restriction is abandoned, forms the subject of a classical research by Stokes* and of many subsequent investigations. The problem is most conveniently treated as one of steady motion. It was pointed out by Rayleighf that if we neglect small quantities of the order a 3 /\ 3 , the solution in the case of infinite depth is contained in the formulae * == - x + ffe^ sin kx, X = - y + $#v cos Jew (1) The equation of the wave-profile (yfr = 0) is found by successive approxima- tions to be y = fie ky cos kx = (1 4- hy -f \k 2 y 2 + . . .) cos kx = P/3 2 + /3 (1 + |A^/3 2 ) cos kx + P/3 2 cos 2kx + p 2 /3 3 cos Skx + ... ; . . .(2) or, if we put /3 (1 + f & 2 /3 2 ) = a, y — \ ka 2 = a cos kx 4- 1 ka 2 cos 2&a> -f f& 2 a 3 cos Skx + ,(3) So far as we have developed it, this coincides with the equation of a trochoid, in which the circumference of the rolling circle is 2ir/k, or \ and the length of the arm of the tracing point is a. We have still to shew that the condition of uniform pressure along this stream-line can be satisfied by a suitably chosen value of c. We have, from (1), without approximation, 2 = const. -gy-\c 2 {\- 2k/3e k v cos kx + k 2 /3 2 e 2k y}, (4) and therefore, at points of the line y — /3e k v cos kx, £ = const. + (kc 2 -g)y- \Wc 2 $ 2 e 2k y r = const. + (kc 2 — g — k 3 c 2 /3 2 ) y+ (5) Hence the condition for a free surface is satisfied, to the present order of approximation, provided ? 2 = | + ^c 2 y8 2 = |(l+A; 2 a 2 ) (6) c< * "On the theory of Oscillatory Waves," Gamb. Trans, viii. (1847) [Papers, i. 197]. The method was one of successive approximation based on the exact equations of Arts. 9 and 20 ante. In a supplement of date 1880 the space-co-ordinates x, y are regarded as functions of the inde- pendent variables <p, \f/ [Papers, i. 314]. t I.e. ante p. 260. The method was subsequently extended so as to include all Stokes' results, Phil. Mag. (6) xxi. 183 [Papers, vi. 11]. 41 8 Surface Waves [chap, ix This determines the velocity of progressive waves of permanent type, and shews that it increases somewhat with the amplitude a. The figure shews the wave-profile, as given by (3), in the case of ha = £, ora/\ = -0796* The approximately trochoidal form gives an outline which is sharper near the crests, and flatter in the troughs, than in the case of the simple-harmonic waves of infinitely small amplitude investigated in Art. 229, and these features become accentuated as the amplitude is increased. If the trochoidal form were exact, instead of merely approximate, the limiting form would have cusps at the crests, as in the case of Gerstner's waves to be considered presently. In the actual problem, which is one of irrotational motion, the extreme form has been shewn by Stokes f, in a very simple manner, to have sharp angles of 120°. The question being still treated as one of steady motion, the motion near the angle will be given by the formulae of Art. 63; viz. if we introduce polar co-ordinates r, 6 with the crest as origin, and the initial line of 6 drawn vertically downwards, we have ty = Cr m cos m6, (7) with the condition that \jr = when 0=±a (say), so that ma — \ir. This formula leads to Q = m Cr™~\ (8) where q is the resultant fluid-velocity. But since the velocity vanishes at the crest, its value at a neighbouring point of the free surface will be given by (f — 2gr cos a, (9) as in Art. 24 (2). Comparing (8) and (9), we see that we must have m = f, and therefore a = \tt\. In the case of progressive waves advancing over still water, the particles at the crests, when these have their extreme forms, are moving forwards with exactly the velocity of the wave. Another point of interest in connection with these waves of permanent type is that they possess, relatively to the undisturbed water, a certain * The approximation in (3) is hardly adequate for so large a value of ka ; see equation (17) below. The figure serves however to indicate the general form of the wave-profile. t Papers, i. 227 (1880). X The wave-profile has been investigated and traced by Michell, "The Highest Waves in Water," Phil. Mag. (5) xxxvi. 430 (1893). He finds that the extreme height is 142 \, and that the wave-velocity is greater than in the case of infinitely small height in the ratio of 1-2 to 1. See also Wilton, Phil. Mag. (6) xxvi. 1053 (1913). 25o] Waves of Finite Height 419 momentum in the direction of wave -propagation. The momentum, per wave- length, of the fluid contained between the free surface and a depth h (beneath the level of the origin), which we will suppose to be great compared with \, is -'/. dxdy = pch\, (10) since yjr = 0, by hypothesis, at the surface, and = ch, by (1), at the great depth h. In the absence of waves, the equation to the upper surface would be y = %ka 2 , by (3), and the corresponding value of the momentum would therefore be pc{h + \ka 2 )X (11) The difference of these results is equal to irpa 2 c, (12) which gives therefore the momentum, per wave-length, of a system of progressive waves of permanent type, moving over water which is at rest at a great depth. To find the vertical distribution of this momentum, we remark that the equation of a stream-line ^ — ch' is found from (2) by writing y-\-ti for y and {3e~ kh ' for (3. The mean-level of this stream-line is therefore given by y=-h' + ik/3 2 e- 2kh ' (13) Hence the momentum, in the case of undisturbed flow, of the stratum of fluid included between the surface and the stream-line in question would be, per wave-length, pc\{h' + P/S a (l-e- M *')} (14) The actual momentum being pch'X, we have, for the momentum of the same stratum in the case of waves advancing over still water, 7rpft 2 c(l-e- 2M ') (15) It appears therefore that the motion of the individual particles, in these progressive waves of permanent type, is not purely oscillatory, and that there is, on the whole, a slow but continued advance in the direction of wave- propagation*. The rate of this flow at a depth K is found approximately by differentiating (15) with respect to h! ', and dividing by p\, viz. it is WaPoer**' (16) This diminishes rapidly from the surface downwards. The further approximation by Stokes, confirmed by the independent cal- culations of Rayleigh and others, gives as the equation of the wave-profile y = const. + a cos hx — {\ka 2 + JjA^a 4 ) cos 2kx + f Pa 3 cos Skx - iPa 4 cos4&#+..., (17) * Stokes, I.e. ante p. 417. Another very simple proof of this statement has been given by Rayleigh, I.e. ante p. 260. 420 Surface Waves [chap, ix with, for the wave-velocity, c 2 = |(l + A; 2 a 2 + J& 4 a 4 +...) (18) A question as to the convergence, both of the series which form the coeffi- cients of the successive cosines when the approximation is continued, and of the resulting series of cosines, was raised by Burnside*, who even expressed a doubt as to the possibility of waves of rigorously permanent type. This led Rayleigh to undertake an extended investigation f, which shewed that the condition of uniformity of pressure at the surface could be satisfied, for sufficiently small values of ha, to a very high degree of accuracy. He inferred that the existence of permanent types up to the highest wave of Michell was practically, if not demonstrably, certain. The existence has at length been definitely established by an investigation of Prof. Levi CivitaJ, which puts an end to an historic controversy. There are one or two simple properties of these permanent waves which come easily from first principles § . The problem being reduced to one of steady motion, let the origin be taken in the mean level, beneath (say) a crest, and let X be the wave-length. Denoting by rj surface-elevation above the mean level, we have, then, ndx=0 (19) /. Also, if q be the surface velocity, and q its value at the mean level, we have q 2 = q 2 -2grj, and therefore / q i dx = q<?\ (20) Jo Again, consider the mass of fluid contained between vertical planes through two successive crests, and bounded below by a plane y= —h x at which the velocity is sensibly horizontal and equal to c. It is easily seen that the total vertical mass-acceleration is zero, since there is no flux of vertical momentum across the boundaries. Hence if p be the surface -pressure, and p x that at the depth h x , / (Pi-P)dx=gpj (h 1 + r } )dx=gph 1 \ (21) But, comparing pressures in the same vertical we have Pi-p=9p(h\+ii)+\{q 2 -<?\ f x and thence / q 2 dx=c 2 X (22) Jo We may express this by saying that the mean square of the surface velocity, per equal increments of #, is equal to c 2 . It follows also from (20) that q = c, i.e. the velocity at the points where the wave-profile meets the mean level is equal to c. * Proc. Lond. Math. Soc. (2) xv. 26 (1916). t Phil. Mag. (6) xxxiii. 381 (1917) [Papers, vi. 478]. $ "Determination rigoureuse des ondes permanentes d'ampleur finie," Math. Ann. xciii. 264 (1925). The extension to waves in a canal of finite depth has been made by Struik, Math. Ann. xcv. 595 (1926). § Levi Civita, I.e. 25(>-25i] Gerstner's Rotational Waves 421 251. A system of exact equations, expressing a possible form of wave- motion when the depth of the fluid is infinite, was given so long ago as 1802 by Gerstner*, and at a later period independently by Rankinef. The circum- stance, however, that the motion in these waves is not irrotational detracts somewhat from the physical interest of the results. If the axis of x be horizontal, and that of y be drawn vertically upwards, the formulae in question may be written x = a + y e*** sin k (a + ct), y = 6— ye kb cos k (a + ct), (1) where the specification is on the Lagrangian plan (Art. 16), viz. a, b are two parameters serving to identify a particle, and x, y are the co-ordinates of this particle at time t The constant k determines the wave-length, and c is the velocity of the waves which are travelling in the direction of ^-negative. To verify this solution, and to determine the value of c, we remark, in the first place, that d(a,b) 6 ' W so that the Lagrangian equation of continuity (Art. 16 (2)) is satisfied. Again, substituting from (1) in the equations of motion (Art. 13), we find jr (- + 9y) = kc 2 e kb sin k (a + ct\ gi (- + 9V) = ~ kc 2 e kb cos k (a + ct) + kc 2 e m ; whence ^ = const. - g \b - t e™> cos k (a + ct) I - c 2 e kb cos k(a + ct) + % c 2 e 2kb . . . .(4) For a particle on the free surface the pressure must be constant; this requires c 2 = g/k, (5) as in Art. 229. This makes V - = const. -#& + ic 2 e 2to (6) It is obvious from (1) that the path of any particle (a, b) is a circle ot radius k' 1 ^. It has already been stated that the motion of the fluid in these waves is rotational. To prove this we remark that = |S{e fc6 sinA;(a + cO} + ce 2fc& 8a, (7~ which is not an exact differential. * Professor of Mathematics at Prague, 1789-1823. His paper, "Theorie der Wellen," was published in the Abh. d. k. bohm. Ges. d. Wiss. 1802 [Gilbert's Annalen d. Physik, xxxii. (1809)]. t "On the Exact Form of Waves near the Surface of Deep Water," Phil. Trans. 1863 [Papers, p. 481], •(3) 422 Surface Waves [chap, ix The circulation in the boundary of the parallelogram whose vertices coincide with the particles (a, b), (a + 8a, b), (a, b + 8b), (a + 8a,b + 8b) is, therefore, - ^ {ce 2kb 8a) 8b, and the area of the circuit is l&^8a8b - (1 - e™) 8a8b. d (a, b) d Hence the vorticity (co) of the element (a, b) is 2kce 2kb W = -]T^ < 8 ) This is greatest at the surface, and diminishes rapidly with increasing depth. Its sense is opposite to that of the revolution of the particles in their circular orbits. A system of waves of the present type cannot therefore be originated from rest, or destroyed, by the action of forces of the kind contemplated in the general theorem of Arts. 17, 33. We may however suppose that by properly adjusted pressures applied to the surface of the waves the liquid is gradually reduced to a state of flow in horizontal lines, in which the velocity (v!) is a function of the ordinate (y f ) only*. In this state we shall have doc' /da = 1, while y' is a function of b determined by the condition <j Wj_ y') d (x, y) . 9 (a, b) d(a,b)' W or l^i-gto (10) This makes % =%%- - 2a> t' = 2te**», (11) cb dy' db db ' and therefore u = ce 2kb (12) Hence, for the genesis of the waves by ordinary forces, we require as a foundation an initial horizontal motion, in the direction opposite to that of propagation of the waves ultimately set up, which diminishes rapidly from the surface downwards, according to the law (12), where b is a function of y' determined by y'^b-lk- 1 ** (13) It is to be noted that these rotational waves, when established, have zero momentum. The figure shews the forms of the lines of equal pressure 6 = const., for a series of equidistant values of 6f. These curves are trochoids, obtained by * For a fuller statement of the argument see Stokes' Papers, i. 222. t The diagram is very similar to the one given originally by Gerstner, and copied more or less closely by subsequent writers. A version of Gerstner's investigation, including in one respect a correction, was given in the second edition of this work, Art. 233. 251-252] Gerstner's Waves 423 rolling circles of radii Ar 1 on the under sides of the lines y=b + kr\ the distances of the tracing points from the respective centres being k" 1 ^. Any one of these lines may be taken as representing the free surface, the extreme admissible form being that of the cycloid. The dotted lines represent the successive forms taken by a line of particles which is vertical when it passes through a crest or a trough. 252. Scott Russell, in his interesting experimental investigations*, was led to pay great attention to a particular type which he called the 'solitary wave.' This is a wave consisting of a single elevation, of height not necessarily small compared with the depth of the fluid, which, if properly started, may travel for a considerable distance along a uniform canal, with little or no change of type. Waves of depression, of similar relative amplitude, were found not to possess the same character of permanence, but to break up into series of shorter waves. Russell's 'solitary' type may be regarded as an extreme case of Stokes' oscillatory waves of permanent type, the wave-length being great compared with the depth of the canal, so that the widely separated elevations are practically independent of one another. The methods of approximation employed by Stokes become, however, unsuitable when the wave-length much exceeds the depth ; and subsequent investigations of solitary waves of permanent type have proceeded on different lines. The first of these was given independently by Boussinesqt and RayleighJ. The latter writer, treating the problem as one of steady motion, starts virtually from the formula d (1) iy <\> + ity = F(x + iy) = e dx F(x), * "Beport on Waves," Brit. Ass. Rep. 1844. t Comvtes Rendus, June 19, 1871. J I.e. ante p. 260. 424 Surface Waves [chap, ix where F(x) is real. This is especially appropriate to cases, such as the present, where one of the family of stream-lines is straight. We derive from (1) 0-y-f^+f^ 1 *-..., ♦-y^'-f^+fl^-..., (2) where the accents denote differentiations with respect to x. The stream-line \^ = here forms the bed of the canal, whilst at the free surface we have \//-= -cA, where c is the uniform velocity, and h the depth, in the parts of the fluid at a distance from the wave, whether in front or behind. The condition of uniform pressure along the free surface gives u 2 + v 2 =c 2 -2g(y-h), (3) or, substituting from (2), F' 2 -y 2 F'F"+y 2 F" 2 +... = c 2 -2g(y-h) (4) But, from (2) we have, along the same surface, yF'-f l F'"+...= -ck (5) It remains to eliminate F between (4) and (5) ; the result will be a differential equation to determine the ordinate y of the free surface. If (as we will suppose) the function F' (x) and its differential coefficients vary so slowly with x that they change only by a small fraction of their values when x increases by an amount comparable with the depth h, the terms in (4) and (5) will be of gradually diminishing magnitude, and the elimination in question can be carried out by a process of successive approximation. Thus, from (5), ^— 7+s^+"— «»jj+«* , G)" + -} ! (6) and if we retain only terms up to the order last written, the equation (4) becomes y 2 3 y \y) ^ \y) h 2 c 2 h 2 ' or, on reduction, 1 2y" \y' 2 _ 1 Zg(y-h) y 2 3 y 3y 2 h 2 c 2 h 2 {n If we multiply by y', and integrate, determining the arbitrary constant so as to make y = for y = k, we obtain 1 \y' 2 _ 1 y-h g(y-h) 2 y 3 y h^ h 2 c 2 h 2 ' or ^fc^-f)- .(8) Hence y' vanishes only for y = h and y=c 2 /g, and since the last factor must be positive, it appears that c 2 \g is a maximum value of y. Hence the wave is necessarily one of eleva- tion only, and, denoting by a the maximum height above the undisturbed level, we have c 2 =g{h + a), (9) which is exactly the empirical formula for the wave-velocity adopted by Russell. The extreme form of the wave must, as in Art. 250, have a sharp crest of 120° ; and since the fluid is there at rest we shall have c 2 = 2ga. If the formula (9) were applicable to such an extreme case, it would follow that a = h. If we put, for shortness, h 2 (h + a) 19 we find, from (8), '-±i0-9*. -<"> 252] Solitary Wave 425 a sech 2 .(12) the integral of which is if the origin of x be taken beneath the summit. There is no definite ' length ' of the wave, but we may note, as a rough indication of its extent, that the elevation has one-tenth of its maximum value when 0/6 = 3*636. \y x' O x The annexed drawing of the curve y = l+^sechH# represents the wave-profile in the case a=\h. For lower waves the scale of y must be contracted, and that of x enlarged, as indicated by the annexed table giving the ratio bjh, which determines the horizontal scale, for various values of a/A. It will be found, on reviewing the above investigation, that the approximations consist in neglecting the fourth power of the ratio (A + a)/2fe*. If we impress on the fluid a velocity — c parallel to x we get the case of a progressive wave on still water. It is not difficult to shew that, if the ratio alh be small, the path of each particle is then an arc of a parabola having its axis vertical and apex upwards f. It might appear, at first sight, that the above theory is inconsistent with the results of Art. 187, where it was argued that a wave of finite height whose length is great compared with the depth must inevitably suffer a continual change of form as it advances, the changes being the more rapid the greater the elevation above the undisturbed level. The investigation referred to postulates, however, a length so great that the vertical acceleration may be neglected, with the result that the horizontal velocity is sensibly uniform from top to bottom (Art. 169). The numerical table above given shews, on the other hand, that the longer the ' solitary wave ' is, the lower it is. In other words, the more nearly it approaches to the character of a ' long ' wave, in the sense of Art. 169, the more easily is the change of type averted by a slight adjustment of the particle- velocities % . The motion at the outskirts of the solitary wave can be represented by a very simple formula. Considering a progressive wave travelling in the direction of ^-positive, and taking the origin in the bottom of the canal, at a point in the front part of the wave, we assume (p = Ae~ m ^- ct ) cos my (13) This satisfies V 2 <£ = 0, and the surface-condition d<j) a/h bjh •1 1-915 •2 1-414 •3 1-202 •4 1-080 •5 1-000 •6 •943 •7 •900 •8 •866 •9 •839 1-0 •816 +9zz = ° dt* ' » dy .(14) * The theory of the solitary wave has been treated by Weinstein, Lincei (6) iii. 463 (1926), by the method of Levi Civita referred to in Art. 250. He finds that the formula (9) is a very close approximation. f Boussinesq, I.e. % Stokes^ "On the Highest Wave of Uniform Propagation," Proc. Camb. Phil. Soc. iv. 361 (1883) [Papers, v. 140]. 426 Surface Waves [chap, ix will also be satisfied for y= h, provided „ , tan mh ° 2 =^ "1ST (15) This will be found to agree approximately with Kayleigh's investigation if we put m=b~ 1 . The above remark, which was communicated to the author by the late Sir George Stokes*, was suggested by an investigation by McCowant, who shewed that the formula izlz^ ^(x+iy)+ata,nh \m(x+iy) (16) satisfies the conditions very approximately, provided c 2 =- tanraA, (17) m and «ia=§sin 2 m (A + §a), a=atan^ra(A + a), (18) where a denotes the maximum elevation above the mean level, and a is a subsidiary constant. In a subsequent paper J the extreme form of the wave when the crest has a sharp angle of 120° was examined. The limiting value of the ratio a/h was found to be '78, in which case the wave-velocity is given by c 2 = 1 'bQgh. 253. By a slight modification the investigation of Rayleigh and Boussinesq can be made to give the theory of a system of oscillatory waves of finite height in a canal of limited depth §. In the steady-motion form of the problem the momentum per wave-length (X) is repre- sented by llpudxdy = -pi I ^dxdy = -p^X, •(19) where ty x corresponds to the free surface. If h be the mean depth, this momentum may be equated to pch\, where c denotes (in a sense) the mean velocity of the stream. On this understanding we have, at the surface, y\r x = —ch, as before. The arbitrary constant in (3), on the other hand, must be left for the moment undetermined, so that we write u 2 + v 2 = C-2gy (20) We then find, in place of (8), y"=^y-i)^-y)(y-h2\ (21) where h x , h 2 are the upper and lower limits of y, and e 2 k 2 l = ~ (22) It is implied that I cannot be greater than h 2 . If we now write y = ^i cos 2 ^ + A 2 sm2 X > (23) we find /3^ = V{l-£ 2 sin 2 x}, (24) * Cf. Papers, v. 62. t " On the Solitary Wave," Phil. Mag. (5) xxxii. 45 (1891). { "On the Highest Wave of Permanent Type," Phil. Mag. (5) xxxviii. 351 (1894). § Korteweg and De Vries, "On the Change of Form of Long Waves advancing in a Beet- angular Canal, and on a New Type of Long Stationary Waves," Phil. Mag. (5) xxxix. 422 (1895). The method adopted by these writers is somewhat different. Moreover, as the title indicates, the paper includes an examination of the manner in which the wave-profile is changing at any instant, if the conditions for permanency of type are not satisfied. For other modifications of Rayleigh 's method reference may be made to Gwyther, Phil. Mag. (5)1. 213, 308, 349 (1900). 252-254] Solitary Wave 427 'V{fi3>}; k2=h -& (26) Hence, if the origin of x be taken at a crest, we have Jo J <?x J(l -k 2 sin 2 x The wave-length is given by -/a^(x.*). ( 26 ) and y=A 2 +(^i-^2)cn 2 |. [mod. &] (27)* ^W wa-^W 2 ^' - (28) Again, from (23) and (24), ^-^ ^S^ ^-yWW+ft-OJkW) (29) Since this must be equal to AX, we have (h-l)F 1 (k) = (h 1 -l)E 1 (k) (30) In equations (25), (28), (30) we have four relations connecting the six quantities A 1} h 2i I, k, X, /3, so that if two of these be assigned the rest are analytically determinate. The wave-velocity c is then given by (22) t. For example, the form of the waves, and their velocity, are determined by the length X, and the height h x of the crests above the bottom. The solitary wave of Art. 252 is included as a particular case. If we put l=k 2 , we have &=1, and the formulae (28) and (30) then shew that X= oo , h 2 = h. 254. The theory of waves of permanent type has been brought into rela- tion with general dynamical principles by Helmholtzj. If in the equations of motion of a 'gyrostatic' system, Art. 141 (23), we put dV dV dV *--£ *"£' •- Q "~£' (1) where V is the potential energy, it appears that the conditions for steady motion, with q 1% q 2> ... q n constant, are 4(F + Z)-0, ^(V + K) = 0, .... £ n (V+mO, ...(2) where K is the energy of the motion corresponding to any given values of the co-ordinates q lt q 2i ... q n when these are prevented from varying by the application of suitable extraneous forces. This energy is here supposed expressed in terms of the constant momenta corresponding to the ignored co-ordinates %, %', ..., and of the palpable co-ordinates q lf q%, ..- q n - It may however also be expressed in terms of the * The waves represented by (27) are called 'cnoidal waves' by the authors cited. For the method of proceeding to a higher approximation we must refer to the original paper. f When the depth is finite, a question arises as to what is meant exactly by the • velocity of propagation.' The velocity adopted in the text is that of the wave-profile relative to the centre of inertia of the mass of fluid included between two vertical planes at a distance apart equal to the wave-length. Cf. Stokes, Papers, i. 202. J "Die Energie der Wogen und des Windes," Berl. Monatsber, July 17, 1890 [Wiss^ Abh. iii. 333] . 428 Surface Waves [chap, ix velocities %, ^', ... and the co-ordinates q lt q 2 , ... q n \ in this form we denote it by Tq. It may be shewn, exactly as in Art. 142, that dT /dq r = — dK/dq r , so that the conditions (2) are equivalent to i {V - T ° )=0 - i (F - r ° )=0 ' -' 4^-^=0. ..,(3) Hence the condition for free steady motion with any assigned constant values of q lt q%, ... q n is that the corresponding value of V + K, or of V — T , should be stationary. Cf. Art. 203 (7). Further, if in the equations of Art. 141 we write — dV/dq r + Q r for Q r , so that Q r now denotes a component of extraneous force, we find, on multiplying by <7i, q 2 , ... q n in order, and adding, ^.(®+ V + K)=Q 1 q 1 +Q 2 q 2 +...+Q n q ni (4) where ® is the part of the energy which involves the velocities q 1} q 2 , • •• qu- it follows, by the same argument as in Art. 205, that the condition for 'secular' stability, when there are dissipative forces affecting the co-ordinates qi, q 2 , ... q n , but not the ignored co-ordinates X> X> -•-> * s tna ^ T^+if should be a minimum. In the application to the problem of stationary waves, it will tend to clearness if we eliminate all infinities from the question by imagining that the fluid circulates in a ring- shaped canal of uniform rectangular section (the sides being horizontal and vertical), of very large radius. The generalized velocity x corresponding to the ignored co-ordinate may be taken to be the flux per unit breadth of the channel, and the constant momentum of the circulation may be replaced by the cyclic constant k. The co-ordinates q u q 2 , ... q n of the general theory are now represented by the value of the surface-elevation (i/) considered as a function of the longitudinal space-co-ordinate x. The corresponding components of extraneous force are represented by arbitrary pressures applied to the surface. If I denote the whole length of the circuit, then considering unit breadth of the canal we have V=\gp^dx, (5) where rj is subject to the condition f rjdx^O (6) If we could with the same ease obtain a general expression for the kinetic energy of the steady motion corresponding to any prescribed form of the surface, the condition in either of the forms above given would, by the usual processes of the Calculus of Varia- tions, lead to a determination of the possible forms, if any, of stationary waves*. For some general considerations bearing on the problem of stationary waves on the common surface of two currents reference may be made to Helmholtz' paper. This also contains, at the end, some speculations, based on calculations of energy and momentum, as to the length of the waves which would be excited in the first instance by a wind of given velocity. These appear to involve the assumption that the waves will necessarily be of permanent type, since it is only on some such hypothesis that we get a determinate value for the momentum of a train of waves of small amplitude. 254-255] Dynamical Condition for Permanent Type 429 Practically, this is not feasible, except by methods of successive approximation, but we may illustrate the question by reproducing, on the basis of the present theory, the results already obtained for 'long' waves of infinitely small amplitude. If h be the depth of the canal, the velocity in any section when the surface is maintained at rest, with arbitrary elevation 77, is x/(k + 17), where % is the flux. Hence, for the cyclic constant, K =*j o v,)-^4*(i+^ o w), ( 7) approximately, where the term of the first order in 77 has been omitted, in virtue of (6). The kinetic energy, ^p<jc> may be expressed in terms of either % or k. We thus obtain the forms I ^* e y( 1+ mj>*)' < 8 > K -i P -¥( l -mfo" 2dx ) (9) The variable part of P— T is ip (g- pj | o V<&, (10) and that of 7+ K is *('-£)&*' (n) It is obvious that these are both stationary for 17 — ; and that they will be stationary for any infinitely small values of 77, provided ^ 2 =^A 3 , or < 2 =ghl 2 . If we put x = c ^i or k = cI, this condition gives c 2 =gh, (12) in agreement with Art. 175. It appears, moreover, that rj = makes V+ K a maximum or a minimum according as c 2 is greater or less than gh. In other words, the plane form of the surface is secularly stable if, and only if, c < *J(gh). It is to be remarked, however, that the dissipative forces here contemplated are of a special character, viz. they affect the vertical motion of the surface, but not (directly) the flow of the liquid. It is otherwise evident from Art. 175 that if pressures be applied to maintain any given constant form of the surface, then if c 2 > gh these pressures must be greatest over the elevations and least over the depressions. Hence if the pressures be removed, the inequalities of the surface will tend to increase. Wave-Propagation in Two Dimensions. 255. We may next consider some cases of wave-propagation in two horizontal dimensions x, y. The axis of z being drawn vertically upwards, we have, on the hypothesis of infinitely small motions, p _ d(p P where cp satisfies V 2 <£ = (2) The arbitrary function F(t) may be supposed merged in the value ofd<f>/dt If the origin be taken in the undisturbed surface, and if f denote the elevation at time t above this level, the pressure-condition to be satisfied at the surface is '-ffl~ <* --gz + F(t), (1) 430 Surface Waves [chap, ix and the kinematical surface-condition is dt [>_Uo' w cf. Art. 227. Hence, for z = 0, we must have -si+tfsE-0, (5) a* 2 r *a* or, in the case of simple-harmonic motion, "> = ^ (6) if the time-factor be e i(Tt+e) . The fluid being supposed to extend to infinity, horizontally and down- wards, we may briefly examine, in the first place, the effect of a local initial disturbance of the surface, in the case of symmetry about the origin. The typical solution for the case of initial rest is easily seen, on reference to Art. 100, to be * = /^Wo(M,j (7) %= cos <rt Jo (kvr), J provided cr 2 = gk y (8) as in Art. 228. To generalize this, subject to the condition of symmetry, we have recourse to the theorem /(«)= rj (kvr)kdk r/(a)J (ka)ada (9) Jo Jo of Art. 100 (12). Thus, corresponding to the initial conditions, r=/(o, tf» =o, (io) .(ii) r<x> gij^ Q.+ roo \ we have ^—g] e kz Jo(k^)kdk\ f(a)J (ka)ada,) J o <r Jo I f— cos at Jo (kvr)kdk I f(a)J (ka)ada. If the initial elevation be concentrated in the immediate neighbourhood of the origin, then, assuming rf(a)2>rrada = l, (12) Jo we have 6 = ^-1 e kz J {k^)kdk ^l3) Ait J a Expanding, and making use of (8), we get 255] Propagation in two Dimensions 431 Ifweput z — -rcos6, «r = rsin0, (15) /•oo \ vfe have e^J (krff)dk = - , (16) Jo r by Art. 102 (9), and thence* ^Ji(far)*-dfc-g)*J-»!^ f (17) where yu. = cos (of. Art. 85). Hence * = 2^ 1^^ 3T 73 + -5l J*—-- J" - (18) From this the value of f is to be obtained by (3). It appears from Arts. 84, 85 that p 2 „ + i(o) = o, fhW-t-)' 1 !"''^ 11 , (19) whence 1 fi»^ V.&fgW. l'.ffl.ffl/gA' { , MV i ?- 2^l2!^ irUJ + 10! W "-}■ - (2 ° )T It follows that any particular phase of the motion is associated with a particular value of gfi/tj, and thence that the various phases travel radially outwards from the origin, each with a constant acceleration. No exact equivalent for (20), analogous to the formula (21) of Art. 238 which was obtained in the two-dimensional form of the problem, and accord- ingly suitable for discussion in the case where gt 2 /^ is large, has been dis- covered. An approximate value may however be obtained by Kelvin's method (Art. 241). Since Jo {z) is a fluctuating function which tends as z increases to have the same period 27r as sin z, the elements of the integral in (13) will for the most part cancel one another with the exception of those for which tdcr\dk=Ts, or km = gt 2 l^i^, (21) nearly. Now when km is large we have Mk&)=tAA sin(& CT + £7r), (22) approximately, by Art. 194 (15), and we may therefore replace (13) by *- <7* 1 \ e kz cos (crt-km-lir) dk (23) r* J Comparing with (7) and (9) of Art. 241, and putting now z = 0, we find as the surface value of <f> » M ig7wg| " K " M (24) * Hobson, Proc. Lond. Math. Soc. xxv. 72, 73 (1893). This formula may, however, be dispensed with ; see the first footnote on p. 385 ante. t This result was given by Cauchy and Poisson. 432 Surface Waves [chap, ix where k and a are to be expressed in terms of cr and t by means of (8) and (21). Note has here been taken of the fact that d 2 a/dk 2 is negative. Since at = (gkt 2 )$ = 2km, td 2 a/dk 2 = - ighk~% = - 2m z \gt 2 , . . .(25) we have <£o=-^ - — sin — (26) r 2%7TV 2 4*7 V ' The surface elevation is then given by (3). Keeping, for consistency, only the most important term, we find f_.jg_oo.tf, ( 27) 2^7TOT 3 4ct which agrees with the result obtained, in other ways, by Cauchy and Poisson. It is not necessary to dwell on the interpretation, which will be readily understood from what has been said in Art. 240 with respect to the two- dimensional case. The consequences were worked out in some detail by Poisson on the hypothesis of an initial paraboloidal depression. When the initial data are of impulse, the typical solution is pcf> = cos at e kz J (km), \ .(28) ?= sin at Jo (km), (" Hi J which, being generalized, gives, for the initial conditions P 4>o = F(m), f-0, (29) the solution d> = P I rco /-co 6 = - cos ate 7 * J (km) kdk F (a) J (ka) a da, P J o Jo ? = a sin at J (km) kdk \ F (a) J (ka)ada. gpJ o Jo ) ...(30) gp In particular, for a concentrated impulse at the origin, such that \ C °F(a)27rada = l, (31) J o 1 f 00 we find £ = 0— cos at e kz Jo(km)kdk (32) lirp J o Since this may be written *-=i- If-— «*••/■.(*»)*# (33) Zirp ot J o 0" we find, performing 1/gp.d/dt on the results contained in (18) and (20), 1 fPjQt) gt 2 2\P 2 („) (gt 2 ) 2 3\P 3 (ri 9 ~~2iro\ r 2 2! r 8 4! r 4 27rp [ r t 2"rrpm - 8 > 5! W 9! \«J '"]' 1- ...(34) 255-256] Travelling Disturbance 433 Again, when igt 2 /^ is large, we have, in place of (27), {>- -j£_Bn*£ (35)* 256. We proceed to consider the effect of a local disturbance of pressure advancing with constant velocity over the surface f. This will give us, at all events as to the main features, an explanation of the peculiar system of waves which is seen to accompany a ship moving through sufficiently deep water. A complete investigation, after the manner of Arts. 242, 243, would be somewhat difficult; but the general characteristics can readily be made out with the help of preceding results, the procedure being similar to that of Art 249. Let us suppose that we have a pressure-point moving with velocity c along the axis of #, in the negative direction, and that at the instant under consideration it has reached the point 0. The elevation f at any point P may be regarded as due to a series of infinitely small impulses applied at equal infinitely short intervals at points of the axis of x to the right of 0. Of the annular wave-systems thus successively generated, those only will combine to produce a sensible effect at P which had their origin in the neighbourhood of certain points Q, which are determined by the consideration that the phase at P is 'stationary' for variations in the position of Q. Now if t is the time which the source of disturbance has taken to travel from Q to 0, the phase of the waves at P, originated at Q, is £+*-. w where ct = QP (Art. 255 (35)). Hence the condition for stationary phase is *=T < 2 > * The waves due to various types of explosive action beneath the surface have been studied by Terazawa, Proc. Roy. Soc. A, xcii. 57 (1915), and by the author of this work, I.e. ante p. 410, and Proc. Lond. Math. Soc. (2) xxi. 359 (1922). t For a more general treatment of such questions reference may be made to a paper by the author, "On Wave-Patterns due to a Travelling Disturbance," Phil. Mag. (6) xxxi. 539 (1916). 434 Surface Waves [chap. IX Since, in this differentiation, and P are regarded as fixed, we have is = c cos 6, where 6 = OQP ; hence 0Q = ct = 2vsece (3) It is further evident that the points in the immediate neighbourhood of P, for which the resultant phase is the same as at P, will lie in a line perpendicular to QP. A glance at the figure on p. 433 then shews that a curve of uniform phase is characterized by the property that the tangent bisects the interval between the origin and the foot of the normal. If p denote the perpendicular from the origin to the tangent, and 6 the angle which p makes with the axis of %, we have, by a known formula, PZ=- whence dp dd ; p = a cos 2 6. .(6) The forms of the curves defined by (5) are shewn in the annexed figure*, which is traced from the equations dt) x=pcosd — ~sinO= J a (5 cos — cos 30), cLu y=psind + ~ cos 6 = — \a (sin V + sin SO), au * Cf. Sir W. Thomson, "On Ship Waves," Proc. Inst. Mech. Eng. Aug. 3, 1887 [Popular Lectures, iii. 482], where a similar drawing is given. The investigation there referred to, based apparently on the theory of 'group-velocity,' was not published. See also E. E. Froude, "On Ship Eesistance," Papers of the Greenock Phil. Soc. Jan. 19, 1894. It is shewn immediately that there is a difference of phase between the two branches meeting at a cusp, so that the drawing does not represent quite accurately the configuration of the wave-ridges. 256] Wave-Pattern 435 The phase-difference from one curve to the corresponding portion of the next is 2tt. This implies a difference 2-7rc 2 /<7 m the parameter a. Since two curves of the above kind pass through any assigned point P within the boundaries of the wave-system, it is evident that there are two corresponding effective positions of Q in the foregoing discussion. These are determined by a very simple construction. If the line OP be bisected in C, and a circle be drawn on GP as diameter, meeting the axis of a; in B, ly R 2 , the perpendiculars PQi, PQ2 to PRi, PR 2 > respectively, will meet the axis in the required points, Q lt Q 2 - For CR X is parallel to PQx and equal to %PQi', the perpendicular from on PR ± produced is therefore equal to PQ\. Similarly, the perpendicular from on PR 2 produced is equal to PQ 2 . The points Q ly Q 2 coincide when OP makes an angle sin -1 J , or 19° 28', with the axis of symmetry. For greater inclinations of OP they are imaginary. It appears also from (6) that the values of x, y are stationary when sin 2 = J ; this gives a series of cusps lying on the straight lines 1 ^= + 2V2 = ± tan 19° 28' (7) To obtain an approximate estimate of the actual height of the waves, in the different parts of the system, we have recourse to the formula (35) of Art. 255. If P denote the total disturbing pressure, the elevation at P due to the annular wave-system started at a point Q to the right of may be written Sr=- o if.-4 -sin^.PoS<, (8) 8V2^- Sm 4^ P »^ where *r=PQ, t = OQ/c. This is to be integrated with respect to t, but (as already explained) the only parts of the integral which contribute appreciably to the final result will be those for which t has very nearly the values (r 1} t 2 ) corresponding to the special points Q 1} Q 2 above mentioned. As regards the phase, we have, writing t = r + t' } 9t\ + t' dt VW J T 1 . 2 Idt 2 VW. + . ...(9) where, in the terms in [ ], t is to be put equal to tx or r 2 as the case may be. 436 Surface Waves [chap, ix The second term vanishes by hypothesis, since the phase at P for waves started near Qi or Q 2 is ' stationary.' Again, we find !?l(9!l\- JL-9L • 9 t2 ( 2 * 2 *\ dt 2 \4m) ~ 2*r *r* V + 4 V *x 3 w *) ' c n c 2 sin 2 . Since <xr = ccos0, ot = , (10) '57 this gives, with the help of (2), [1(g)] -£<*-*■*<> <"> Owing to the fluctuations of the trigonometrical term no great error will be committed if we neglect the variation of the first factor in (8), or if, further, we take the limits of integration with respect to t' to be ± oo . We have then, approximately, f=-8^/>(£>--<")* •00 f^n- 2 J&/X&+-"')* < 12 > 8 \J2TTplZ2 where mj 2 = /- (i _ tan 2 X \ m 2 2 = ^- (tan 2 <9 2 - J), (13) ZtZTj Z'33'2 and the suffixes refer to the points Q 1} Q 2 of the last figure. Since (°° cos m¥W==P° sin rnH'Ht' = */(&ir)lm, (14) J — 00 •/ —00 where the positive value of m is understood, we find f_ ^fo .sin(^ +i7r ) gZggo ,sin(fA_ i7r ). 8^2^*^^1*011 V4^i / 8V2ir*psr,*m s V 4ct 2 / (15) The two terms give the parts due to the transverse and lateral waves respectively. Since ^ x = PQ X = Jcri cos lf ct 2 = PQ2 — icr 2 cos 2 , it appears that if we consider either term by itself, the phase is constant along the corresponding part of the curve p = <oy = acos 2 0, whilst the elevation varies as V^iPo sec*fl 7rV 3 a*' \/|l-3sin 2 <9| At the cusps, where the two systems combine, there is a phase-difference of a quarter-period between them. The formulae make f infinite at a cusp, where sin 2 #= J, but this is merely an indication of the failure of our approximation. That the elevation at a point P in the neighbourhood of a cusp would be relatively great might have been foreseen, since, as appears from (9) and (11), the range of points on 256-256 a] Ship Waves . 437 the axis of a; which have sent waves to P in sensibly the same phase is then abnormally extended. The infinity which occurs when 6 = \tt is of a some- what different character, being due to the artificial nature of the assumption we have made, of a pressure concentrated at a point. With a diffused pressure this difficulty would disappear*. It is to be noticed, moreover, that the whole of this investigation applies only to points for which gt 2 /^ is large ; cf. Arts. 240, 255. It will be found on examination that this restriction is equivalent to an assumption that the parameter a is large compared with 27rc 2 /g. The argument therefore does not apply without reserve to the parts of the wave-pattern near the origin. 256 a. As already indicated, wave-systems of the above type are generated by other forms of travelling disturbance. Some of these cases are amenable to calculation. The translation of a submerged sphere, for instance, has been dealt with by Havelockf, and the wave-resistance determined. The writer J has discussed by another method the translation of a submerged solid, without restriction as to its precise form or orientation. The results are naturally simplest when the direction of motion coincides with one of the three directions of 'permanent translation' considered in Art. 124. The resistance is then given by the formula B _££±jWl (17) 7T/3C 6 v y Here A denotes the appropriate inertia- coefficient from Art. 121, Q is the volume of the solid, c its velocity, and I=[*\ec 5 0e-W! c2 - seG * e dd y (18) Jo where / is the depth of immersion. Another form of this integral (due to Havelock) is / = ie-"|&fo(«)+(l + ^) #i(<*)}, (19) in the accepted notation of Bessel Functions §, with a = gf/c 2 . For a sphere we have A = § 7rpa?, Q = f 7ra 3 , where a is the radius. Hence if M ' be the mass of fluid displaced, R = m'g.(fy 3 (yff.I, (20)|| in agreement with Havelock's result. As an example, if c = \/(gf), R = -365 M'g (a//) 8 . * More elaborate investigations have been carried out by Hopf in a dissertation of date Munich, 1909, and Hogner, Arkiv for Matem. xvii. (1923). The latter writer examines in particular the shape of the waves near the 'cusps,' where the two systems cross. t Proc. Roy. Soc. A, xciii. 520 (1917); xcv. 354 (1918). See also Green, Phil. Mag. (6) xxxvi. 48 (1918). % Proc. Boy. Soc. A, cxi. 14 (1926). § Watson, p. 172. II This formula was given incorrectly in the author's paper. 438 Surface Waves [chap, ix A graph of R as a function of c is given by Havelock; it has a general resemblance to the curve on p. 416. In a subsequent paper* the same method is applied by Havelock to a travelling disturbance consisting of various arrangements of (double) sources, with important applications to the wave-resistance of ships. Some further reference to the theoretical literature of wave-resistance may be in place here. Although the mode of disturbance is different, the action of the bows of a ship may be compared to that of a pressure-point. The diagram on p. 434 accounts for the two systems of transverse and lateral waves which are observed, and for the especially con- spicuous 'echelon' waves near the cusps, where the two systems cross. If in addition we imagine a negative pressure -point at the stern we get a rough representation of the action of the ship as a whole. With varying speeds the stern waves may tend partially to annul, or to reinforce, the effect of the bow waves, with the result that the resistance may be expected to fluctuate up and down as the length of the ship is increased, or the speed varied f. It is found in fact that the curve of resistance as a function of the speed exhibits several maxima (or 'humps') with the corresponding minima, as well as a general increase. To obtain an improved representation of what happens in the immediate neighbourhood of a ship and to calculate the consequent resistance is of course a difficult matter, but attempts have been made with considerable success. A beginning was made by J. H. Michelle with an idealized ship form, which differs mainly from that of a real ship in that the incliuation of the surface to the medial plane is everywhere small. This plan has recently been followed up by Wigley §, who has discussed a variety of forms (subject to the same limitation), calculated their resistance, and compared it with the results of model experiments, with a considerable measure of qualitative agreement. Havelock, in a long series of papers || has discussed the effect of various features in the design of a ship, such as length of 'parallel middle body,' mean draught, and so on. His method consists (in part) in the choice of a suitable arrangement of travelling sources, and is accordingly free from the special restriction above mentioned IT. A general formula for the wave-resistance of geometrically similar bodies, similarly immersed (wholly or partially), was given long ago by Froude. Since the resistance can only depend on the speed, the density of the fluid, the intensity of gravity, and on some linear magnitude which fixes the scale, considerations of dimensions shew that it must satisfy a relation of the form R=p^fi$). (2D where c is the speed, and I the characteristic linear magnitude. It will be * Proc. Roy. Soc. A, cxviii. 24 (1927). t W. Froude, "On the Effect on the Wave-Making Eesistance of Ships of Length of Parallel Middle Body," Trans. Inst. Nav. Arch. xvii. (1877). AlsoR. E. Froude, " On the Leading Phenomena of the Wave-Making Resistance of Ships," Trans. Inst. Nav. Arch. xxii. (1881), where drawings of actual wave-patterns under varied conditions of speed are given, which are, as to their main features, in striking agreement with the results of the above theory. Some of these drawings are reproduced in Kelvin's paper in the Proc. Inst. Mech. Eng. above cited. J Phil. Mag. (5) xlv. 106 (1898). § Trans. Inst. Nav. Arch, lxviii. 124 (1926); lxix. 27 (1927); lxxii. (1930). || 'In the Proc. Roy. Soc. from 1909 onwards. T[ Excellent accounts of the development of the subject are given by Hogner, Proc. Congress. App. Math. Delft, 1924, p. 146, and Wigley, Congress for techn. Mechanics, Stockolm, 1930. 256 a-256 b] Effect of Limited Depth 439 noticed that (17) is a particular case of this. It follows from (21) that the wave-resistance of a ship can be inferred from a model experiment provided the value of l/c 2 is the same on the model as on the full scale. 256 b. To examine the modification produced in the wave-pattern when the depth of the water has to be taken into account, the argument on p. 433 must be put in a more general form. If, as before, t is the time the pressure- point has taken to travel from Q to 0, it may be shewn that the phase of the disturbance at P, due to the impulse delivered at Q, will differ only by a constant from k(Vt-n), (22) where 27r/k is the predominant wave-length in the neighbourhood of P, and V the corresponding wave- velocity *. This predominant wave-length is determined by the condition that the phase is stationary for variations of the wave-length only, i.e. ^r.k(Vt-vf) = O t or w=Ut, (23) where U, = d(kV)/dk, is the group- velocity (Art. 236). For the effective part of the disturbance at P, the phase (22) must further be stationary as regards variations in the position of Q; hence, differentiating partially with respect to t, we have bt = F, or F=ccos0, (24) since is = c cos 0. Now, referring to the figure on p. 433, we have p = ct cos 6 — nr = Vt — ts (25) Hence for a given wave-ridge p will bear a constant ratio to the wave- length X, and in passing from one wave-ridge to the next this ratio will increase (or decrease) by unity. Since \ is determined as a function of 6 by (24), this gives the relation between p and 0. Thus in the case of infinite depth, the formula (24) gives c 2 cos 2 <9 = F 2 = |^, (26) and the required relation is of the form p = acos 2 d, (27) as above. When the depth (h) is finite, we have c 2 cos 2 = F 2 = ^tanh^, (28) Lit A, and the relation is ^tanh- = -%os 2 0, (29) a p gh x * The symbol c, which was previously employed in this sense, now denotes the velocity of the pressure-point over the water. 440 Surface Waves [chap, ix where the values of a for successive wave-ridges are in arithmetic progression. Since the expression on the left-hand side cannot exceed unity, it appears that if c 2 > gh there will be an inferior limit to the value of 6, determined by cos 2 d = gh/c 2 , (30) the curve then extending to infinity. It follows that when the speed of the disturbing influence exceeds \/'{gh) the transverse waves disappear, and we have only the lateral waves. This tends to diminish the wave-making resistance (cf. Art. 249)*. The changes in the configuration of the wave-pattern as the ratio c 2 /gh increases from zero to infinity are traced by Havelockf. Standing Waves in Limited Masses of Water. 257. The problem of free oscillations in two horizontal dimensions (%, y), in the case where the depth is uniform and the fluid is bounded laterally by vertical walls, can be reduced to the same analytical form as in Art. 190. If the origin be taken in the undisturbed surface, and if f denote the elevation at time t above this level, the conditions to be satisfied at the free surface are as in Art. 255 (3), (4). The equation of continuity, V 2 $ = 0, and the condition of zero vertical motion at the depth z — — h, are both satisfied by <p = fa cosh k(z + h), (1) where fa is a function of x, y, such that t&+t£+**-° (2 > The form of fa and the admissible values of k are determined by this equation, and by the condition that &-* (*> at the vertical walls. The corresponding values of the ' speed ' (<r) of the oscillations are then given by the surface-condition (6), of Art. 255 ; viz. we have a 2 = gh tanh Jch (4) This makes f = — si